Properties

Label 32-1260e16-1.1-c1e16-0-6
Degree $32$
Conductor $4.036\times 10^{49}$
Sign $1$
Analytic cond. $1.10245\times 10^{16}$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 4·7-s − 8·14-s + 2·16-s + 24·19-s − 8·25-s + 4·28-s − 16·29-s − 8·31-s + 24·37-s − 48·38-s + 16·47-s + 16·50-s + 32·53-s + 32·58-s + 8·59-s + 16·62-s − 6·64-s − 48·74-s + 24·76-s + 8·83-s − 32·94-s − 8·100-s − 8·103-s − 64·106-s + 8·109-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.51·7-s − 2.13·14-s + 1/2·16-s + 5.50·19-s − 8/5·25-s + 0.755·28-s − 2.97·29-s − 1.43·31-s + 3.94·37-s − 7.78·38-s + 2.33·47-s + 2.26·50-s + 4.39·53-s + 4.20·58-s + 1.04·59-s + 2.03·62-s − 3/4·64-s − 5.57·74-s + 2.75·76-s + 0.878·83-s − 3.30·94-s − 4/5·100-s − 0.788·103-s − 6.21·106-s + 0.766·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(1.10245\times 10^{16}\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1260} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(20.73415791\)
\(L(\frac12)\) \(\approx\) \(20.73415791\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T + 3 T^{2} + p^{2} T^{3} + 3 T^{4} - p T^{5} - 7 T^{6} - 3 p^{2} T^{7} - 7 p^{2} T^{8} - 3 p^{3} T^{9} - 7 p^{2} T^{10} - p^{4} T^{11} + 3 p^{4} T^{12} + p^{7} T^{13} + 3 p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \)
3 \( 1 \)
5 \( ( 1 + T^{2} )^{8} \)
7 \( 1 - 4 T + 16 T^{2} - 68 T^{3} + 188 T^{4} - 468 T^{5} + 1328 T^{6} - 3284 T^{7} + 6854 T^{8} - 3284 p T^{9} + 1328 p^{2} T^{10} - 468 p^{3} T^{11} + 188 p^{4} T^{12} - 68 p^{5} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \)
good11 \( 1 - 68 T^{2} + 2180 T^{4} - 46812 T^{6} + 798068 T^{8} - 11533812 T^{10} + 145791676 T^{12} - 1699254444 T^{14} + 19011779286 T^{16} - 1699254444 p^{2} T^{18} + 145791676 p^{4} T^{20} - 11533812 p^{6} T^{22} + 798068 p^{8} T^{24} - 46812 p^{10} T^{26} + 2180 p^{12} T^{28} - 68 p^{14} T^{30} + p^{16} T^{32} \)
13 \( 1 - 100 T^{2} + 5348 T^{4} - 198140 T^{6} + 5637172 T^{8} - 129840148 T^{10} + 2495923420 T^{12} - 40770066124 T^{14} + 571339339990 T^{16} - 40770066124 p^{2} T^{18} + 2495923420 p^{4} T^{20} - 129840148 p^{6} T^{22} + 5637172 p^{8} T^{24} - 198140 p^{10} T^{26} + 5348 p^{12} T^{28} - 100 p^{14} T^{30} + p^{16} T^{32} \)
17 \( 1 - 168 T^{2} + 13880 T^{4} - 751096 T^{6} + 29923612 T^{8} - 935973160 T^{10} + 23941747080 T^{12} - 514532623928 T^{14} + 9440913455942 T^{16} - 514532623928 p^{2} T^{18} + 23941747080 p^{4} T^{20} - 935973160 p^{6} T^{22} + 29923612 p^{8} T^{24} - 751096 p^{10} T^{26} + 13880 p^{12} T^{28} - 168 p^{14} T^{30} + p^{16} T^{32} \)
19 \( ( 1 - 12 T + 158 T^{2} - 1308 T^{3} + 10440 T^{4} - 66012 T^{5} + 391538 T^{6} - 1971116 T^{7} + 9248974 T^{8} - 1971116 p T^{9} + 391538 p^{2} T^{10} - 66012 p^{3} T^{11} + 10440 p^{4} T^{12} - 1308 p^{5} T^{13} + 158 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
23 \( 1 - 176 T^{2} + 15448 T^{4} - 913680 T^{6} + 41467228 T^{8} - 1550033328 T^{10} + 49589827176 T^{12} - 1383348399760 T^{14} + 33907824809542 T^{16} - 1383348399760 p^{2} T^{18} + 49589827176 p^{4} T^{20} - 1550033328 p^{6} T^{22} + 41467228 p^{8} T^{24} - 913680 p^{10} T^{26} + 15448 p^{12} T^{28} - 176 p^{14} T^{30} + p^{16} T^{32} \)
29 \( ( 1 + 8 T + 100 T^{2} + 360 T^{3} + 3380 T^{4} + 3064 T^{5} + 62140 T^{6} - 238824 T^{7} + 906806 T^{8} - 238824 p T^{9} + 62140 p^{2} T^{10} + 3064 p^{3} T^{11} + 3380 p^{4} T^{12} + 360 p^{5} T^{13} + 100 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 4 T + 98 T^{2} + 516 T^{3} + 5624 T^{4} + 35252 T^{5} + 225022 T^{6} + 1588916 T^{7} + 7569102 T^{8} + 1588916 p T^{9} + 225022 p^{2} T^{10} + 35252 p^{3} T^{11} + 5624 p^{4} T^{12} + 516 p^{5} T^{13} + 98 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 12 T + 188 T^{2} - 1604 T^{3} + 16708 T^{4} - 119548 T^{5} + 988388 T^{6} - 6004372 T^{7} + 1131038 p T^{8} - 6004372 p T^{9} + 988388 p^{2} T^{10} - 119548 p^{3} T^{11} + 16708 p^{4} T^{12} - 1604 p^{5} T^{13} + 188 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( 1 - 184 T^{2} + 20472 T^{4} - 1751720 T^{6} + 121699868 T^{8} - 7243988536 T^{10} + 380165048008 T^{12} - 17911615742824 T^{14} + 768666981378246 T^{16} - 17911615742824 p^{2} T^{18} + 380165048008 p^{4} T^{20} - 7243988536 p^{6} T^{22} + 121699868 p^{8} T^{24} - 1751720 p^{10} T^{26} + 20472 p^{12} T^{28} - 184 p^{14} T^{30} + p^{16} T^{32} \)
43 \( 1 - 352 T^{2} + 65784 T^{4} - 8492960 T^{6} + 836964316 T^{8} - 66242040800 T^{10} + 4333734314568 T^{12} - 238312249642272 T^{14} + 11112687909569926 T^{16} - 238312249642272 p^{2} T^{18} + 4333734314568 p^{4} T^{20} - 66242040800 p^{6} T^{22} + 836964316 p^{8} T^{24} - 8492960 p^{10} T^{26} + 65784 p^{12} T^{28} - 352 p^{14} T^{30} + p^{16} T^{32} \)
47 \( ( 1 - 8 T + 200 T^{2} - 1480 T^{3} + 21484 T^{4} - 142376 T^{5} + 1546616 T^{6} - 9131112 T^{7} + 83479334 T^{8} - 9131112 p T^{9} + 1546616 p^{2} T^{10} - 142376 p^{3} T^{11} + 21484 p^{4} T^{12} - 1480 p^{5} T^{13} + 200 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 16 T + 6 p T^{2} - 3232 T^{3} + 39656 T^{4} - 299616 T^{5} + 2879090 T^{6} - 18163120 T^{7} + 161612366 T^{8} - 18163120 p T^{9} + 2879090 p^{2} T^{10} - 299616 p^{3} T^{11} + 39656 p^{4} T^{12} - 3232 p^{5} T^{13} + 6 p^{7} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( ( 1 - 4 T + 216 T^{2} - 212 T^{3} + 21484 T^{4} + 35116 T^{5} + 1537192 T^{6} + 4969884 T^{7} + 95982982 T^{8} + 4969884 p T^{9} + 1537192 p^{2} T^{10} + 35116 p^{3} T^{11} + 21484 p^{4} T^{12} - 212 p^{5} T^{13} + 216 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
61 \( 1 - 424 T^{2} + 85912 T^{4} - 11360056 T^{6} + 1147532060 T^{8} - 97928338472 T^{10} + 7427530642088 T^{12} - 510650158156408 T^{14} + 32321138943506822 T^{16} - 510650158156408 p^{2} T^{18} + 7427530642088 p^{4} T^{20} - 97928338472 p^{6} T^{22} + 1147532060 p^{8} T^{24} - 11360056 p^{10} T^{26} + 85912 p^{12} T^{28} - 424 p^{14} T^{30} + p^{16} T^{32} \)
67 \( 1 - 568 T^{2} + 166008 T^{4} - 32876264 T^{6} + 4920317404 T^{8} - 589683946296 T^{10} + 58602724633032 T^{12} - 4934611468716584 T^{14} + 356328084895246342 T^{16} - 4934611468716584 p^{2} T^{18} + 58602724633032 p^{4} T^{20} - 589683946296 p^{6} T^{22} + 4920317404 p^{8} T^{24} - 32876264 p^{10} T^{26} + 166008 p^{12} T^{28} - 568 p^{14} T^{30} + p^{16} T^{32} \)
71 \( 1 - 444 T^{2} + 111268 T^{4} - 19597764 T^{6} + 2684602292 T^{8} - 301990393612 T^{10} + 28967151924188 T^{12} - 2434802829480948 T^{14} + 182479018420426774 T^{16} - 2434802829480948 p^{2} T^{18} + 28967151924188 p^{4} T^{20} - 301990393612 p^{6} T^{22} + 2684602292 p^{8} T^{24} - 19597764 p^{10} T^{26} + 111268 p^{12} T^{28} - 444 p^{14} T^{30} + p^{16} T^{32} \)
73 \( 1 - 708 T^{2} + 243620 T^{4} - 54480956 T^{6} + 8938972724 T^{8} - 1152660018548 T^{10} + 122264196408284 T^{12} - 11016340703744524 T^{14} + 860580405090458646 T^{16} - 11016340703744524 p^{2} T^{18} + 122264196408284 p^{4} T^{20} - 1152660018548 p^{6} T^{22} + 8938972724 p^{8} T^{24} - 54480956 p^{10} T^{26} + 243620 p^{12} T^{28} - 708 p^{14} T^{30} + p^{16} T^{32} \)
79 \( 1 - 776 T^{2} + 298264 T^{4} - 75873816 T^{6} + 14350843100 T^{8} - 2143612302472 T^{10} + 261769978394792 T^{12} - 26666593413127256 T^{14} + 2290442673574914502 T^{16} - 26666593413127256 p^{2} T^{18} + 261769978394792 p^{4} T^{20} - 2143612302472 p^{6} T^{22} + 14350843100 p^{8} T^{24} - 75873816 p^{10} T^{26} + 298264 p^{12} T^{28} - 776 p^{14} T^{30} + p^{16} T^{32} \)
83 \( ( 1 - 4 T + 196 T^{2} + 268 T^{3} + 25748 T^{4} + 39276 T^{5} + 3021596 T^{6} + 7672636 T^{7} + 246037366 T^{8} + 7672636 p T^{9} + 3021596 p^{2} T^{10} + 39276 p^{3} T^{11} + 25748 p^{4} T^{12} + 268 p^{5} T^{13} + 196 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
89 \( 1 - 864 T^{2} + 376504 T^{4} - 109545120 T^{6} + 23755420700 T^{8} - 4063039933920 T^{10} + 6361453333320 p T^{12} - 65504477701979168 T^{14} + 6355980862575907782 T^{16} - 65504477701979168 p^{2} T^{18} + 6361453333320 p^{5} T^{20} - 4063039933920 p^{6} T^{22} + 23755420700 p^{8} T^{24} - 109545120 p^{10} T^{26} + 376504 p^{12} T^{28} - 864 p^{14} T^{30} + p^{16} T^{32} \)
97 \( 1 - 708 T^{2} + 266468 T^{4} - 70482684 T^{6} + 14473556148 T^{8} - 2426815167348 T^{10} + 342304459764508 T^{12} - 41313766425365708 T^{14} + 4305828967872497814 T^{16} - 41313766425365708 p^{2} T^{18} + 342304459764508 p^{4} T^{20} - 2426815167348 p^{6} T^{22} + 14473556148 p^{8} T^{24} - 70482684 p^{10} T^{26} + 266468 p^{12} T^{28} - 708 p^{14} T^{30} + p^{16} T^{32} \)
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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

−2.45369962033876810413811874945, −2.35965208929934620948642652453, −2.31159566732134880403207509600, −2.12726098354186889570762968682, −2.04085604592119426002799405134, −1.92893408703537665012021644186, −1.91513878378910681799511781183, −1.91068930049639494765294222666, −1.86800363458700432789316751669, −1.79024571694650763305685482426, −1.72948575443409595195449672474, −1.65806669874442034437050456647, −1.51107008978138246506374436588, −1.50765809878986385831296269855, −1.11679643586909628987565259826, −1.01729481619968072130910797233, −1.00081360878366170367159832714, −0.997558202580968321330242214000, −0.876863490092731376518864127700, −0.813144125612419738781755113393, −0.793440863809752224318764430463, −0.60680449142134696130237027535, −0.55370687474428030260207714205, −0.31923766445610927112640705460, −0.22513323127364435222849359682, 0.22513323127364435222849359682, 0.31923766445610927112640705460, 0.55370687474428030260207714205, 0.60680449142134696130237027535, 0.793440863809752224318764430463, 0.813144125612419738781755113393, 0.876863490092731376518864127700, 0.997558202580968321330242214000, 1.00081360878366170367159832714, 1.01729481619968072130910797233, 1.11679643586909628987565259826, 1.50765809878986385831296269855, 1.51107008978138246506374436588, 1.65806669874442034437050456647, 1.72948575443409595195449672474, 1.79024571694650763305685482426, 1.86800363458700432789316751669, 1.91068930049639494765294222666, 1.91513878378910681799511781183, 1.92893408703537665012021644186, 2.04085604592119426002799405134, 2.12726098354186889570762968682, 2.31159566732134880403207509600, 2.35965208929934620948642652453, 2.45369962033876810413811874945

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

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