Properties

Label 36-845e18-1.1-c1e18-0-3
Degree $36$
Conductor $4.824\times 10^{52}$
Sign $1$
Analytic cond. $8.40250\times 10^{14}$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 14·3-s + 4-s + 87·9-s + 14·12-s + 2·16-s + 2·17-s − 28·23-s − 9·25-s + 304·27-s + 24·29-s + 87·36-s − 78·43-s + 28·48-s + 47·49-s + 28·51-s − 16·53-s − 6·61-s + 15·64-s + 2·68-s − 392·69-s − 126·75-s + 78·79-s + 583·81-s + 336·87-s − 28·92-s − 9·100-s + 50·101-s + ⋯
L(s)  = 1  + 8.08·3-s + 1/2·4-s + 29·9-s + 4.04·12-s + 1/2·16-s + 0.485·17-s − 5.83·23-s − 9/5·25-s + 58.5·27-s + 4.45·29-s + 29/2·36-s − 11.8·43-s + 4.04·48-s + 47/7·49-s + 3.92·51-s − 2.19·53-s − 0.768·61-s + 15/8·64-s + 0.242·68-s − 47.1·69-s − 14.5·75-s + 8.77·79-s + 64.7·81-s + 36.0·87-s − 2.91·92-s − 0.899·100-s + 4.97·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{18} \cdot 13^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(36\)
Conductor: \(5^{18} \cdot 13^{36}\)
Sign: $1$
Analytic conductor: \(8.40250\times 10^{14}\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((36,\ 5^{18} \cdot 13^{36} ,\ ( \ : [1/2]^{18} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(763.8224029\)
\(L(\frac12)\) \(\approx\) \(763.8224029\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( ( 1 + T^{2} )^{9} \)
13 \( 1 \)
good2 \( 1 - T^{2} - T^{4} - 3 p^{2} T^{6} + 15 T^{8} - p^{2} T^{10} + 49 T^{12} - 5 p^{5} T^{14} + 3 p^{6} T^{16} + 177 T^{18} + 3 p^{8} T^{20} - 5 p^{9} T^{22} + 49 p^{6} T^{24} - p^{10} T^{26} + 15 p^{10} T^{28} - 3 p^{14} T^{30} - p^{14} T^{32} - p^{16} T^{34} + p^{18} T^{36} \)
3 \( ( 1 - 7 T + 10 p T^{2} - 32 p T^{3} + 257 T^{4} - 199 p T^{5} + 1270 T^{6} - 2515 T^{7} + 1576 p T^{8} - 8395 T^{9} + 1576 p^{2} T^{10} - 2515 p^{2} T^{11} + 1270 p^{3} T^{12} - 199 p^{5} T^{13} + 257 p^{5} T^{14} - 32 p^{7} T^{15} + 10 p^{8} T^{16} - 7 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
7 \( 1 - 47 T^{2} + 1222 T^{4} - 22786 T^{6} + 335731 T^{8} - 4116765 T^{10} + 43379820 T^{12} - 8187525 p^{2} T^{14} + 3301528296 T^{16} - 24369707501 T^{18} + 3301528296 p^{2} T^{20} - 8187525 p^{6} T^{22} + 43379820 p^{6} T^{24} - 4116765 p^{8} T^{26} + 335731 p^{10} T^{28} - 22786 p^{12} T^{30} + 1222 p^{14} T^{32} - 47 p^{16} T^{34} + p^{18} T^{36} \)
11 \( 1 - 93 T^{2} + 4363 T^{4} - 135323 T^{6} + 3097806 T^{8} - 55713058 T^{10} + 825007658 T^{12} - 10535620709 T^{14} + 122336384660 T^{16} - 1359715540834 T^{18} + 122336384660 p^{2} T^{20} - 10535620709 p^{4} T^{22} + 825007658 p^{6} T^{24} - 55713058 p^{8} T^{26} + 3097806 p^{10} T^{28} - 135323 p^{12} T^{30} + 4363 p^{14} T^{32} - 93 p^{16} T^{34} + p^{18} T^{36} \)
17 \( ( 1 - T + 73 T^{2} - 5 p T^{3} + 2536 T^{4} - 3330 T^{5} + 3336 p T^{6} - 84603 T^{7} + 1001750 T^{8} - 1626586 T^{9} + 1001750 p T^{10} - 84603 p^{2} T^{11} + 3336 p^{4} T^{12} - 3330 p^{4} T^{13} + 2536 p^{5} T^{14} - 5 p^{7} T^{15} + 73 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18} )^{2} \)
19 \( 1 - 164 T^{2} + 13554 T^{4} - 760833 T^{6} + 32844155 T^{8} - 1162812786 T^{10} + 34972124545 T^{12} - 911637838879 T^{14} + 20846665057897 T^{16} - 421008872847156 T^{18} + 20846665057897 p^{2} T^{20} - 911637838879 p^{4} T^{22} + 34972124545 p^{6} T^{24} - 1162812786 p^{8} T^{26} + 32844155 p^{10} T^{28} - 760833 p^{12} T^{30} + 13554 p^{14} T^{32} - 164 p^{16} T^{34} + p^{18} T^{36} \)
23 \( ( 1 + 14 T + 187 T^{2} + 65 p T^{3} + 11643 T^{4} + 65013 T^{5} + 369694 T^{6} + 1590535 T^{7} + 7973892 T^{8} + 32968331 T^{9} + 7973892 p T^{10} + 1590535 p^{2} T^{11} + 369694 p^{3} T^{12} + 65013 p^{4} T^{13} + 11643 p^{5} T^{14} + 65 p^{7} T^{15} + 187 p^{7} T^{16} + 14 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
29 \( ( 1 - 12 T + 239 T^{2} - 2089 T^{3} + 23847 T^{4} - 165119 T^{5} + 1395032 T^{6} - 8044995 T^{7} + 55530902 T^{8} - 273802049 T^{9} + 55530902 p T^{10} - 8044995 p^{2} T^{11} + 1395032 p^{3} T^{12} - 165119 p^{4} T^{13} + 23847 p^{5} T^{14} - 2089 p^{6} T^{15} + 239 p^{7} T^{16} - 12 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
31 \( 1 - 289 T^{2} + 41527 T^{4} - 3965691 T^{6} + 283713610 T^{8} - 16265661666 T^{10} + 781111061326 T^{12} - 32408447119893 T^{14} + 1186670838892784 T^{16} - 38832125250494682 T^{18} + 1186670838892784 p^{2} T^{20} - 32408447119893 p^{4} T^{22} + 781111061326 p^{6} T^{24} - 16265661666 p^{8} T^{26} + 283713610 p^{10} T^{28} - 3965691 p^{12} T^{30} + 41527 p^{14} T^{32} - 289 p^{16} T^{34} + p^{18} T^{36} \)
37 \( 1 - 215 T^{2} + 27956 T^{4} - 2637689 T^{6} + 200377937 T^{8} - 12786724230 T^{10} + 707186145651 T^{12} - 34460408008583 T^{14} + 1498224782515471 T^{16} - 58441324013438438 T^{18} + 1498224782515471 p^{2} T^{20} - 34460408008583 p^{4} T^{22} + 707186145651 p^{6} T^{24} - 12786724230 p^{8} T^{26} + 200377937 p^{10} T^{28} - 2637689 p^{12} T^{30} + 27956 p^{14} T^{32} - 215 p^{16} T^{34} + p^{18} T^{36} \)
41 \( 1 - 334 T^{2} + 58563 T^{4} - 7133321 T^{6} + 673115143 T^{8} - 52062396777 T^{10} + 3411402596700 T^{12} - 193218997898893 T^{14} + 9575273063401150 T^{16} - 417904622456524335 T^{18} + 9575273063401150 p^{2} T^{20} - 193218997898893 p^{4} T^{22} + 3411402596700 p^{6} T^{24} - 52062396777 p^{8} T^{26} + 673115143 p^{10} T^{28} - 7133321 p^{12} T^{30} + 58563 p^{14} T^{32} - 334 p^{16} T^{34} + p^{18} T^{36} \)
43 \( ( 1 + 39 T + 992 T^{2} + 18178 T^{3} + 268333 T^{4} + 3283897 T^{5} + 34393990 T^{6} + 311846483 T^{7} + 2477363362 T^{8} + 17298907959 T^{9} + 2477363362 p T^{10} + 311846483 p^{2} T^{11} + 34393990 p^{3} T^{12} + 3283897 p^{4} T^{13} + 268333 p^{5} T^{14} + 18178 p^{6} T^{15} + 992 p^{7} T^{16} + 39 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
47 \( 1 - 10 p T^{2} + 109335 T^{4} - 16976837 T^{6} + 1990283131 T^{8} - 187734448461 T^{10} + 14768030059516 T^{12} - 990064529508401 T^{14} + 57318404440641010 T^{16} - 2885746816454283939 T^{18} + 57318404440641010 p^{2} T^{20} - 990064529508401 p^{4} T^{22} + 14768030059516 p^{6} T^{24} - 187734448461 p^{8} T^{26} + 1990283131 p^{10} T^{28} - 16976837 p^{12} T^{30} + 109335 p^{14} T^{32} - 10 p^{17} T^{34} + p^{18} T^{36} \)
53 \( ( 1 + 8 T + 214 T^{2} + 2003 T^{3} + 24407 T^{4} + 220058 T^{5} + 1980433 T^{6} + 286737 p T^{7} + 128187853 T^{8} + 840286876 T^{9} + 128187853 p T^{10} + 286737 p^{3} T^{11} + 1980433 p^{3} T^{12} + 220058 p^{4} T^{13} + 24407 p^{5} T^{14} + 2003 p^{6} T^{15} + 214 p^{7} T^{16} + 8 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
59 \( 1 - 571 T^{2} + 156884 T^{4} - 27430013 T^{6} + 3407252233 T^{8} - 318601069374 T^{10} + 23317433164299 T^{12} - 1396007831606787 T^{14} + 74491137286929183 T^{16} - 4115895955388638254 T^{18} + 74491137286929183 p^{2} T^{20} - 1396007831606787 p^{4} T^{22} + 23317433164299 p^{6} T^{24} - 318601069374 p^{8} T^{26} + 3407252233 p^{10} T^{28} - 27430013 p^{12} T^{30} + 156884 p^{14} T^{32} - 571 p^{16} T^{34} + p^{18} T^{36} \)
61 \( ( 1 + 3 T + 390 T^{2} + 1184 T^{3} + 74287 T^{4} + 216023 T^{5} + 9008768 T^{6} + 23900771 T^{7} + 762120612 T^{8} + 1762483611 T^{9} + 762120612 p T^{10} + 23900771 p^{2} T^{11} + 9008768 p^{3} T^{12} + 216023 p^{4} T^{13} + 74287 p^{5} T^{14} + 1184 p^{6} T^{15} + 390 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
67 \( 1 - 785 T^{2} + 303403 T^{4} - 76890748 T^{6} + 14350888243 T^{8} - 2099933841489 T^{10} + 250325888459264 T^{12} - 24920313634739285 T^{14} + 2104897664631485584 T^{16} - \)\(15\!\cdots\!63\)\( T^{18} + 2104897664631485584 p^{2} T^{20} - 24920313634739285 p^{4} T^{22} + 250325888459264 p^{6} T^{24} - 2099933841489 p^{8} T^{26} + 14350888243 p^{10} T^{28} - 76890748 p^{12} T^{30} + 303403 p^{14} T^{32} - 785 p^{16} T^{34} + p^{18} T^{36} \)
71 \( 1 - 825 T^{2} + 342167 T^{4} - 94289339 T^{6} + 19275597610 T^{8} - 3096252795458 T^{10} + 404234373355630 T^{12} - 43799465466314549 T^{14} + 3989123734741178944 T^{16} - \)\(30\!\cdots\!66\)\( T^{18} + 3989123734741178944 p^{2} T^{20} - 43799465466314549 p^{4} T^{22} + 404234373355630 p^{6} T^{24} - 3096252795458 p^{8} T^{26} + 19275597610 p^{10} T^{28} - 94289339 p^{12} T^{30} + 342167 p^{14} T^{32} - 825 p^{16} T^{34} + p^{18} T^{36} \)
73 \( 1 - 802 T^{2} + 316393 T^{4} - 82094448 T^{6} + 15798632052 T^{8} - 2407691443640 T^{10} + 302329513287236 T^{12} - 32056752858359824 T^{14} + 2912640674000078702 T^{16} - \)\(22\!\cdots\!60\)\( T^{18} + 2912640674000078702 p^{2} T^{20} - 32056752858359824 p^{4} T^{22} + 302329513287236 p^{6} T^{24} - 2407691443640 p^{8} T^{26} + 15798632052 p^{10} T^{28} - 82094448 p^{12} T^{30} + 316393 p^{14} T^{32} - 802 p^{16} T^{34} + p^{18} T^{36} \)
79 \( ( 1 - 39 T + 1259 T^{2} - 27763 T^{3} + 530120 T^{4} - 8248342 T^{5} + 114036272 T^{6} - 1354582637 T^{7} + 14478789698 T^{8} - 135406639814 T^{9} + 14478789698 p T^{10} - 1354582637 p^{2} T^{11} + 114036272 p^{3} T^{12} - 8248342 p^{4} T^{13} + 530120 p^{5} T^{14} - 27763 p^{6} T^{15} + 1259 p^{7} T^{16} - 39 p^{8} T^{17} + p^{9} T^{18} )^{2} \)
83 \( 1 - 895 T^{2} + 410202 T^{4} - 126424102 T^{6} + 29174564143 T^{8} - 5330886286713 T^{10} + 797078717299648 T^{12} - 99525056653618305 T^{14} + 10507025951118018420 T^{16} - \)\(94\!\cdots\!61\)\( T^{18} + 10507025951118018420 p^{2} T^{20} - 99525056653618305 p^{4} T^{22} + 797078717299648 p^{6} T^{24} - 5330886286713 p^{8} T^{26} + 29174564143 p^{10} T^{28} - 126424102 p^{12} T^{30} + 410202 p^{14} T^{32} - 895 p^{16} T^{34} + p^{18} T^{36} \)
89 \( 1 - 1093 T^{2} + 579151 T^{4} - 198776416 T^{6} + 49824362039 T^{8} - 9744228955297 T^{10} + 1549075508043372 T^{12} - 205467100962513797 T^{14} + 23102939204412290600 T^{16} - \)\(22\!\cdots\!87\)\( T^{18} + 23102939204412290600 p^{2} T^{20} - 205467100962513797 p^{4} T^{22} + 1549075508043372 p^{6} T^{24} - 9744228955297 p^{8} T^{26} + 49824362039 p^{10} T^{28} - 198776416 p^{12} T^{30} + 579151 p^{14} T^{32} - 1093 p^{16} T^{34} + p^{18} T^{36} \)
97 \( 1 - 604 T^{2} + 172238 T^{4} - 31674589 T^{6} + 4561719807 T^{8} - 604802336474 T^{10} + 77459000325861 T^{12} - 9061162614902931 T^{14} + 946144848956219789 T^{16} - 92795798238373469620 T^{18} + 946144848956219789 p^{2} T^{20} - 9061162614902931 p^{4} T^{22} + 77459000325861 p^{6} T^{24} - 604802336474 p^{8} T^{26} + 4561719807 p^{10} T^{28} - 31674589 p^{12} T^{30} + 172238 p^{14} T^{32} - 604 p^{16} T^{34} + p^{18} T^{36} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.44532033105077874416999174756, −2.37913355051274459718087735365, −2.34903168027664040392753990977, −2.33565902381253114532755148197, −2.30369687894783428480446189376, −2.12993695833205244828867798773, −1.98255837207307797673504128723, −1.97021588517625936823118546930, −1.89513597880046272074423992635, −1.89255753316345585249668830437, −1.87108933717430820141218749045, −1.82180602978540702714043220704, −1.81539040232377144613672871919, −1.78559658907062708249585339687, −1.48742495522701676805496851997, −1.45874850848498383932699729454, −1.42452879126866446919506900343, −1.25341889123585495711550298723, −0.902868092855062256895046412658, −0.73686898895001015142228184997, −0.66157579030691434203374443346, −0.58171822242016605161102487638, −0.54487493305297188036147423628, −0.49984631933051119318606233075, −0.31350793163323435624718218800, 0.31350793163323435624718218800, 0.49984631933051119318606233075, 0.54487493305297188036147423628, 0.58171822242016605161102487638, 0.66157579030691434203374443346, 0.73686898895001015142228184997, 0.902868092855062256895046412658, 1.25341889123585495711550298723, 1.42452879126866446919506900343, 1.45874850848498383932699729454, 1.48742495522701676805496851997, 1.78559658907062708249585339687, 1.81539040232377144613672871919, 1.82180602978540702714043220704, 1.87108933717430820141218749045, 1.89255753316345585249668830437, 1.89513597880046272074423992635, 1.97021588517625936823118546930, 1.98255837207307797673504128723, 2.12993695833205244828867798773, 2.30369687894783428480446189376, 2.33565902381253114532755148197, 2.34903168027664040392753990977, 2.37913355051274459718087735365, 2.44532033105077874416999174756

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.