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Properties

Label 2-12-12.11-c7-0-9
Degree $2$
Conductor $12$
Sign $-0.0253 + 0.999i$
Analytic cond. $3.74862$
Root an. cond. $1.93613$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.29 + 11.0i)2-s + (−17.5 − 43.3i)3-s + (−117. − 50.9i)4-s − 151. i·5-s + (520. − 94.3i)6-s − 893. i·7-s + (834. − 1.18e3i)8-s + (−1.57e3 + 1.51e3i)9-s + (1.68e3 + 349. i)10-s − 7.69e3·11-s + (−151. + 5.98e3i)12-s + 5.58e3·13-s + (9.89e3 + 2.05e3i)14-s + (−6.58e3 + 2.66e3i)15-s + (1.11e4 + 1.19e4i)16-s − 3.13e3i·17-s + ⋯
L(s)  = 1  + (−0.203 + 0.979i)2-s + (−0.374 − 0.927i)3-s + (−0.917 − 0.397i)4-s − 0.543i·5-s + (0.983 − 0.178i)6-s − 0.984i·7-s + (0.576 − 0.817i)8-s + (−0.719 + 0.694i)9-s + (0.532 + 0.110i)10-s − 1.74·11-s + (−0.0253 + 0.999i)12-s + 0.704·13-s + (0.963 + 0.200i)14-s + (−0.503 + 0.203i)15-s + (0.683 + 0.730i)16-s − 0.155i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0253 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0253 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.0253 + 0.999i$
Analytic conductor: \(3.74862\)
Root analytic conductor: \(1.93613\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :7/2),\ -0.0253 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.501040 - 0.513908i\)
\(L(\frac12)\) \(\approx\) \(0.501040 - 0.513908i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.29 - 11.0i)T \)
3 \( 1 + (17.5 + 43.3i)T \)
good5 \( 1 + 151. iT - 7.81e4T^{2} \)
7 \( 1 + 893. iT - 8.23e5T^{2} \)
11 \( 1 + 7.69e3T + 1.94e7T^{2} \)
13 \( 1 - 5.58e3T + 6.27e7T^{2} \)
17 \( 1 + 3.13e3iT - 4.10e8T^{2} \)
19 \( 1 + 1.71e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.78e4T + 3.40e9T^{2} \)
29 \( 1 + 2.04e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.28e5iT - 2.75e10T^{2} \)
37 \( 1 + 3.24e5T + 9.49e10T^{2} \)
41 \( 1 + 3.22e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.18e5iT - 2.71e11T^{2} \)
47 \( 1 + 6.54e5T + 5.06e11T^{2} \)
53 \( 1 - 5.06e4iT - 1.17e12T^{2} \)
59 \( 1 - 1.93e6T + 2.48e12T^{2} \)
61 \( 1 - 1.37e6T + 3.14e12T^{2} \)
67 \( 1 + 2.50e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.68e5T + 9.09e12T^{2} \)
73 \( 1 + 3.41e5T + 1.10e13T^{2} \)
79 \( 1 - 3.19e6iT - 1.92e13T^{2} \)
83 \( 1 + 2.25e6T + 2.71e13T^{2} \)
89 \( 1 - 4.66e6iT - 4.42e13T^{2} \)
97 \( 1 + 5.57e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.98687101291007753958670930474, −16.94930618438966073458234290952, −15.75710830119435704795884998385, −13.71531503004120731485082082798, −12.97724224880488090225057247771, −10.62419582505482561230447834936, −8.350783025181844883130460889016, −7.06290325708507056494841362074, −5.19781598998420420516834956457, −0.57268461090912883470649677242, 3.01022716062072110820166711668, 5.28246757714233321359983068434, 8.620257697020202861689405376855, 10.24275245121854192593082114084, 11.23652469945850990187887528528, 12.83746877545048726078740315049, 14.79579502001833095220741466211, 16.18249006266253341715407057287, 17.97895462054235266028857947605, 18.76284262346782235190141391346

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