Properties

Label 12-2e30-1.1-c7e6-0-0
Degree $12$
Conductor $1073741824$
Sign $1$
Analytic cond. $997794.$
Root an. cond. $3.16169$
Motivic weight $7$
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  + 688·7-s + 5.10e3·9-s + 1.45e3·17-s + 1.29e3·23-s + 2.14e5·25-s + 8.92e4·31-s + 5.21e5·41-s − 1.56e6·47-s − 2.48e6·49-s + 3.51e6·63-s + 7.59e6·71-s + 2.08e6·73-s − 1.60e7·79-s + 1.12e7·81-s + 2.16e6·89-s − 1.08e6·97-s + 6.64e7·103-s − 4.62e6·113-s + 9.98e5·119-s + 6.46e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.40e6·153-s + ⋯
L(s)  = 1  + 0.758·7-s + 2.33·9-s + 0.0716·17-s + 0.0222·23-s + 2.74·25-s + 0.538·31-s + 1.18·41-s − 2.20·47-s − 3.02·49-s + 1.76·63-s + 2.51·71-s + 0.628·73-s − 3.65·79-s + 2.34·81-s + 0.326·89-s − 0.121·97-s + 5.99·103-s − 0.301·113-s + 0.0543·119-s + 3.31·121-s + 0.167·153-s + 0.0168·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{30}\)
Sign: $1$
Analytic conductor: \(997794.\)
Root analytic conductor: \(3.16169\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{32} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{30} ,\ ( \ : [7/2]^{6} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(10.62333989\)
\(L(\frac12)\) \(\approx\) \(10.62333989\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 5102 T^{2} + 547829 p^{3} T^{4} - 397562596 p^{4} T^{6} + 547829 p^{17} T^{8} - 5102 p^{28} T^{10} + p^{42} T^{12} \)
5 \( 1 - 214718 T^{2} + 1269826431 p^{2} T^{4} - 4608734769796 p^{4} T^{6} + 1269826431 p^{16} T^{8} - 214718 p^{28} T^{10} + p^{42} T^{12} \)
7 \( ( 1 - 344 T + 1422245 T^{2} - 124668368 T^{3} + 1422245 p^{7} T^{4} - 344 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
11 \( 1 - 64629022 T^{2} + 2363172524483783 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + 2363172524483783 p^{14} T^{8} - 64629022 p^{28} T^{10} + p^{42} T^{12} \)
13 \( 1 - 205410958 T^{2} + 1687860473809811 p T^{4} - \)\(16\!\cdots\!64\)\( T^{6} + 1687860473809811 p^{15} T^{8} - 205410958 p^{28} T^{10} + p^{42} T^{12} \)
17 \( ( 1 - 726 T + 323301023 T^{2} - 9708009717300 T^{3} + 323301023 p^{7} T^{4} - 726 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
19 \( 1 - 2002416334 T^{2} + 2872909317854295863 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + 2872909317854295863 p^{14} T^{8} - 2002416334 p^{28} T^{10} + p^{42} T^{12} \)
23 \( ( 1 - 648 T + 9708521717 T^{2} - 6547475963760 T^{3} + 9708521717 p^{7} T^{4} - 648 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
29 \( 1 - 47836636078 T^{2} + \)\(14\!\cdots\!79\)\( T^{4} - \)\(30\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!79\)\( p^{14} T^{8} - 47836636078 p^{28} T^{10} + p^{42} T^{12} \)
31 \( ( 1 - 1440 p T + 48354349725 T^{2} - 4324137771289408 T^{3} + 48354349725 p^{7} T^{4} - 1440 p^{15} T^{5} + p^{21} T^{6} )^{2} \)
37 \( 1 - 78937168126 T^{2} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!51\)\( p^{14} T^{8} - 78937168126 p^{28} T^{10} + p^{42} T^{12} \)
41 \( ( 1 - 260622 T + 351195126263 T^{2} - 83834535574878564 T^{3} + 351195126263 p^{7} T^{4} - 260622 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
43 \( 1 - 1505999929054 T^{2} + \)\(97\!\cdots\!71\)\( T^{4} - \)\(34\!\cdots\!32\)\( T^{6} + \)\(97\!\cdots\!71\)\( p^{14} T^{8} - 1505999929054 p^{28} T^{10} + p^{42} T^{12} \)
47 \( ( 1 + 783216 T + 1470820452333 T^{2} + 778339576797138720 T^{3} + 1470820452333 p^{7} T^{4} + 783216 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
53 \( 1 - 3131635055902 T^{2} + \)\(64\!\cdots\!63\)\( T^{4} - \)\(86\!\cdots\!56\)\( T^{6} + \)\(64\!\cdots\!63\)\( p^{14} T^{8} - 3131635055902 p^{28} T^{10} + p^{42} T^{12} \)
59 \( 1 - 8312323804862 T^{2} + \)\(38\!\cdots\!03\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(38\!\cdots\!03\)\( p^{14} T^{8} - 8312323804862 p^{28} T^{10} + p^{42} T^{12} \)
61 \( 1 - 14350504934382 T^{2} + \)\(96\!\cdots\!03\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{6} + \)\(96\!\cdots\!03\)\( p^{14} T^{8} - 14350504934382 p^{28} T^{10} + p^{42} T^{12} \)
67 \( 1 - 35072892237678 T^{2} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{6} + \)\(51\!\cdots\!43\)\( p^{14} T^{8} - 35072892237678 p^{28} T^{10} + p^{42} T^{12} \)
71 \( ( 1 - 3798552 T + 17820666583269 T^{2} - 72937737977373055056 T^{3} + 17820666583269 p^{7} T^{4} - 3798552 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
73 \( ( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} + 23207418111255 p^{7} T^{4} - 1044782 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
79 \( ( 1 + 8007952 T + 49828330384013 T^{2} + \)\(25\!\cdots\!96\)\( T^{3} + 49828330384013 p^{7} T^{4} + 8007952 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
83 \( 1 - 124932236904014 T^{2} + \)\(70\!\cdots\!51\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(70\!\cdots\!51\)\( p^{14} T^{8} - 124932236904014 p^{28} T^{10} + p^{42} T^{12} \)
89 \( ( 1 - 1084542 T + 130289056617383 T^{2} - 94766843887733093316 T^{3} + 130289056617383 p^{7} T^{4} - 1084542 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
97 \( ( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} + 184934786492783 p^{7} T^{4} + 544154 p^{14} T^{5} + p^{21} T^{6} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.169044837161255900095474211360, −7.962540343840501559336118560231, −7.57580825931589858403040245539, −7.23449504136212369919170148786, −7.21701212440006962384602812542, −6.67674772481399100335420585473, −6.65888563982908288301448129517, −6.46629586391283543646346646067, −6.09447505341967614494552090934, −5.67346551674896208644101448274, −5.12002923692211082562771860120, −4.97523076659672317495670699754, −4.91564704537888073003004333394, −4.39242488855139775717314783538, −4.30330922035465681614105333630, −4.03800118799020079338147600589, −3.23214193338561395400116646262, −3.17791790466598936175492726342, −2.98366391105781296884562579858, −2.01549198704809618541833142767, −2.00927200636710120099327563999, −1.47233357333067625863462355272, −1.24965527033562442954500987866, −0.69962857083990405606003934653, −0.51457261522676747015917469618, 0.51457261522676747015917469618, 0.69962857083990405606003934653, 1.24965527033562442954500987866, 1.47233357333067625863462355272, 2.00927200636710120099327563999, 2.01549198704809618541833142767, 2.98366391105781296884562579858, 3.17791790466598936175492726342, 3.23214193338561395400116646262, 4.03800118799020079338147600589, 4.30330922035465681614105333630, 4.39242488855139775717314783538, 4.91564704537888073003004333394, 4.97523076659672317495670699754, 5.12002923692211082562771860120, 5.67346551674896208644101448274, 6.09447505341967614494552090934, 6.46629586391283543646346646067, 6.65888563982908288301448129517, 6.67674772481399100335420585473, 7.21701212440006962384602812542, 7.23449504136212369919170148786, 7.57580825931589858403040245539, 7.962540343840501559336118560231, 8.169044837161255900095474211360

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

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