Properties

Label 32-252e16-1.1-c1e16-0-2
Degree $32$
Conductor $2.645\times 10^{38}$
Sign $1$
Analytic cond. $72250.7$
Root an. cond. $1.41853$
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  − 7-s + 6·11-s + 6·23-s + 16·25-s − 12·29-s + 4·37-s + 4·43-s − 2·49-s + 14·67-s − 6·77-s + 20·79-s − 9·81-s − 20·109-s + 90·113-s − 25·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 6·161-s + 163-s + 167-s − 56·169-s + 173-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.80·11-s + 1.25·23-s + 16/5·25-s − 2.22·29-s + 0.657·37-s + 0.609·43-s − 2/7·49-s + 1.71·67-s − 0.683·77-s + 2.25·79-s − 81-s − 1.91·109-s + 8.46·113-s − 2.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 0.472·161-s + 0.0783·163-s + 0.0773·167-s − 4.30·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(72250.7\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(6.259213755\)
\(L(\frac12)\) \(\approx\) \(6.259213755\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + p^{2} T^{4} - 5 p^{2} T^{6} - 2 p^{2} T^{8} - 5 p^{4} T^{10} + p^{6} T^{12} + p^{8} T^{16} \)
7 \( 1 + T + 3 T^{2} + 20 T^{3} + 53 T^{4} + 3 p T^{5} - 74 T^{6} + 355 T^{7} - 1314 T^{8} + 355 p T^{9} - 74 p^{2} T^{10} + 3 p^{4} T^{11} + 53 p^{4} T^{12} + 20 p^{5} T^{13} + 3 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \)
good5 \( 1 - 16 T^{2} + 159 T^{4} - 998 T^{6} + 4298 T^{8} - 12066 T^{10} + 21316 T^{12} - 64633 T^{14} + 271566 T^{16} - 64633 p^{2} T^{18} + 21316 p^{4} T^{20} - 12066 p^{6} T^{22} + 4298 p^{8} T^{24} - 998 p^{10} T^{26} + 159 p^{12} T^{28} - 16 p^{14} T^{30} + p^{16} T^{32} \)
11 \( ( 1 - 3 T + 26 T^{2} - 69 T^{3} + 265 T^{4} - 756 T^{5} + 235 p T^{6} - 8259 T^{7} + 34846 T^{8} - 8259 p T^{9} + 235 p^{3} T^{10} - 756 p^{3} T^{11} + 265 p^{4} T^{12} - 69 p^{5} T^{13} + 26 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
13 \( 1 + 56 T^{2} + 1590 T^{4} + 26848 T^{6} + 274865 T^{8} + 1615656 T^{10} + 12863830 T^{12} + 363405056 T^{14} + 6601006116 T^{16} + 363405056 p^{2} T^{18} + 12863830 p^{4} T^{20} + 1615656 p^{6} T^{22} + 274865 p^{8} T^{24} + 26848 p^{10} T^{26} + 1590 p^{12} T^{28} + 56 p^{14} T^{30} + p^{16} T^{32} \)
17 \( ( 1 + 58 T^{2} + 1603 T^{4} + 30013 T^{6} + 497674 T^{8} + 30013 p^{2} T^{10} + 1603 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( ( 1 - 77 T^{2} + 3034 T^{4} - 84368 T^{6} + 1811980 T^{8} - 84368 p^{2} T^{10} + 3034 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 3 T + 56 T^{2} - 159 T^{3} + 1321 T^{4} - 3510 T^{5} + 30863 T^{6} - 67173 T^{7} + 842170 T^{8} - 67173 p T^{9} + 30863 p^{2} T^{10} - 3510 p^{3} T^{11} + 1321 p^{4} T^{12} - 159 p^{5} T^{13} + 56 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 + 6 T + 53 T^{2} + 246 T^{3} + 20 p T^{4} + 540 T^{5} - 22066 T^{6} - 299625 T^{7} - 1268282 T^{8} - 299625 p T^{9} - 22066 p^{2} T^{10} + 540 p^{3} T^{11} + 20 p^{5} T^{12} + 246 p^{5} T^{13} + 53 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( 1 + 176 T^{2} + 16407 T^{4} + 1045570 T^{6} + 50906954 T^{8} + 2036295270 T^{10} + 71233973380 T^{12} + 2306080860179 T^{14} + 72156636517806 T^{16} + 2306080860179 p^{2} T^{18} + 71233973380 p^{4} T^{20} + 2036295270 p^{6} T^{22} + 50906954 p^{8} T^{24} + 1045570 p^{10} T^{26} + 16407 p^{12} T^{28} + 176 p^{14} T^{30} + p^{16} T^{32} \)
37 \( ( 1 - T + 82 T^{2} - 88 T^{3} + 3940 T^{4} - 88 p T^{5} + 82 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{4} \)
41 \( 1 - 151 T^{2} + 9564 T^{4} - 353897 T^{6} + 12352577 T^{8} - 588962616 T^{10} + 30503470855 T^{12} - 1573469011633 T^{14} + 72313421958936 T^{16} - 1573469011633 p^{2} T^{18} + 30503470855 p^{4} T^{20} - 588962616 p^{6} T^{22} + 12352577 p^{8} T^{24} - 353897 p^{10} T^{26} + 9564 p^{12} T^{28} - 151 p^{14} T^{30} + p^{16} T^{32} \)
43 \( ( 1 - 2 T - 93 T^{2} - 64 T^{3} + 92 p T^{4} + 12192 T^{5} - 158990 T^{6} - 321485 T^{7} + 7972668 T^{8} - 321485 p T^{9} - 158990 p^{2} T^{10} + 12192 p^{3} T^{11} + 92 p^{5} T^{12} - 64 p^{5} T^{13} - 93 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 - 154 T^{2} + 7683 T^{4} - 245888 T^{6} + 25028264 T^{8} - 1538143278 T^{10} + 38475997042 T^{12} - 55371690107 p T^{14} + 205422570296046 T^{16} - 55371690107 p^{3} T^{18} + 38475997042 p^{4} T^{20} - 1538143278 p^{6} T^{22} + 25028264 p^{8} T^{24} - 245888 p^{10} T^{26} + 7683 p^{12} T^{28} - 154 p^{14} T^{30} + p^{16} T^{32} \)
53 \( ( 1 - 10 T^{2} + 3133 T^{4} + 54785 T^{6} + 5778574 T^{8} + 54785 p^{2} T^{10} + 3133 p^{4} T^{12} - 10 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
59 \( 1 - 376 T^{2} + 75453 T^{4} - 10695122 T^{6} + 1193881406 T^{8} - 111003832932 T^{10} + 8882998756942 T^{12} - 624280670070511 T^{14} + 38988958528434078 T^{16} - 624280670070511 p^{2} T^{18} + 8882998756942 p^{4} T^{20} - 111003832932 p^{6} T^{22} + 1193881406 p^{8} T^{24} - 10695122 p^{10} T^{26} + 75453 p^{12} T^{28} - 376 p^{14} T^{30} + p^{16} T^{32} \)
61 \( 1 + 137 T^{2} + 306 T^{4} - 368483 T^{6} + 33183791 T^{8} + 2225199582 T^{10} - 143899132121 T^{12} + 1205057298539 T^{14} + 1133086682080440 T^{16} + 1205057298539 p^{2} T^{18} - 143899132121 p^{4} T^{20} + 2225199582 p^{6} T^{22} + 33183791 p^{8} T^{24} - 368483 p^{10} T^{26} + 306 p^{12} T^{28} + 137 p^{14} T^{30} + p^{16} T^{32} \)
67 \( ( 1 - 7 T - 108 T^{2} - 293 T^{3} + 11753 T^{4} + 50778 T^{5} - 198827 T^{6} - 2620663 T^{7} - 10701486 T^{8} - 2620663 p T^{9} - 198827 p^{2} T^{10} + 50778 p^{3} T^{11} + 11753 p^{4} T^{12} - 293 p^{5} T^{13} - 108 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 - 361 T^{2} + 64216 T^{4} - 7488268 T^{6} + 623512600 T^{8} - 7488268 p^{2} T^{10} + 64216 p^{4} T^{12} - 361 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
73 \( ( 1 - 341 T^{2} + 62398 T^{4} - 7562288 T^{6} + 647645452 T^{8} - 7562288 p^{2} T^{10} + 62398 p^{4} T^{12} - 341 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 10 T - 123 T^{2} + 844 T^{3} + 11042 T^{4} + 2166 T^{5} - 1268066 T^{6} - 531391 T^{7} + 106354368 T^{8} - 531391 p T^{9} - 1268066 p^{2} T^{10} + 2166 p^{3} T^{11} + 11042 p^{4} T^{12} + 844 p^{5} T^{13} - 123 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
83 \( 1 - 397 T^{2} + 87990 T^{4} - 12501089 T^{6} + 1185510017 T^{8} - 57837981066 T^{10} - 2507933243939 T^{12} + 849901181197043 T^{14} - 92913215010718536 T^{16} + 849901181197043 p^{2} T^{18} - 2507933243939 p^{4} T^{20} - 57837981066 p^{6} T^{22} + 1185510017 p^{8} T^{24} - 12501089 p^{10} T^{26} + 87990 p^{12} T^{28} - 397 p^{14} T^{30} + p^{16} T^{32} \)
89 \( ( 1 + 64 T^{2} + 9487 T^{4} + 662335 T^{6} + 143358802 T^{8} + 662335 p^{2} T^{10} + 9487 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( 1 + 404 T^{2} + 81609 T^{4} + 10963666 T^{6} + 1046175986 T^{8} + 591335472 p T^{10} - 1663618077194 T^{12} - 785812116013063 T^{14} - 99406691908304130 T^{16} - 785812116013063 p^{2} T^{18} - 1663618077194 p^{4} T^{20} + 591335472 p^{7} T^{22} + 1046175986 p^{8} T^{24} + 10963666 p^{10} T^{26} + 81609 p^{12} T^{28} + 404 p^{14} T^{30} + p^{16} T^{32} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.49244566045704957023980481559, −3.48145167061054345163772896916, −3.39159804466724778781459583111, −3.37504205564567990401870780498, −3.04824934444759588219839545704, −2.93295995310648152396624011023, −2.80936715308775075091279209950, −2.77135647220126501795769156536, −2.70636470832980959822848747470, −2.69279369168447787110256183879, −2.56728299692620831787556529294, −2.48395784552566416092576018592, −2.25700964571370554301348390799, −2.10689603886908677230204034580, −2.01122453503060948485626927505, −1.76591330468639726777043987032, −1.73423419363807500988602705783, −1.73112103955815533998682179815, −1.64437967367432163672492386897, −1.49932994233771599537261324510, −0.959044660620802922282608852693, −0.939931289483233095480732753721, −0.894737869254929797507512698193, −0.850831287254953838118291700032, −0.43494931597950584778927546313, 0.43494931597950584778927546313, 0.850831287254953838118291700032, 0.894737869254929797507512698193, 0.939931289483233095480732753721, 0.959044660620802922282608852693, 1.49932994233771599537261324510, 1.64437967367432163672492386897, 1.73112103955815533998682179815, 1.73423419363807500988602705783, 1.76591330468639726777043987032, 2.01122453503060948485626927505, 2.10689603886908677230204034580, 2.25700964571370554301348390799, 2.48395784552566416092576018592, 2.56728299692620831787556529294, 2.69279369168447787110256183879, 2.70636470832980959822848747470, 2.77135647220126501795769156536, 2.80936715308775075091279209950, 2.93295995310648152396624011023, 3.04824934444759588219839545704, 3.37504205564567990401870780498, 3.39159804466724778781459583111, 3.48145167061054345163772896916, 3.49244566045704957023980481559

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

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