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Properties

Label 2-12-12.11-c11-0-15
Degree $2$
Conductor $12$
Sign $-0.245 + 0.969i$
Analytic cond. $9.22011$
Root an. cond. $3.03646$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (41.7 − 17.5i)2-s + (−364. + 210. i)3-s + (1.43e3 − 1.46e3i)4-s − 505. i·5-s + (−1.14e4 + 1.51e4i)6-s − 7.75e4i·7-s + (3.38e4 − 8.62e4i)8-s + (8.82e4 − 1.53e5i)9-s + (−8.88e3 − 2.10e4i)10-s − 7.31e5·11-s + (−2.12e5 + 8.35e5i)12-s + 1.14e6·13-s + (−1.36e6 − 3.23e6i)14-s + (1.06e5 + 1.84e5i)15-s + (−1.03e5 − 4.19e6i)16-s − 3.04e6i·17-s + ⋯
L(s)  = 1  + (0.921 − 0.388i)2-s + (−0.865 + 0.500i)3-s + (0.698 − 0.715i)4-s − 0.0723i·5-s + (−0.603 + 0.797i)6-s − 1.74i·7-s + (0.365 − 0.930i)8-s + (0.498 − 0.866i)9-s + (−0.0280 − 0.0666i)10-s − 1.36·11-s + (−0.245 + 0.969i)12-s + 0.853·13-s + (−0.677 − 1.60i)14-s + (0.0362 + 0.0625i)15-s + (−0.0247 − 0.999i)16-s − 0.519i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(9.22011\)
Root analytic conductor: \(3.03646\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :11/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.18639 - 1.52503i\)
\(L(\frac12)\) \(\approx\) \(1.18639 - 1.52503i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-41.7 + 17.5i)T \)
3 \( 1 + (364. - 210. i)T \)
good5 \( 1 + 505. iT - 4.88e7T^{2} \)
7 \( 1 + 7.75e4iT - 1.97e9T^{2} \)
11 \( 1 + 7.31e5T + 2.85e11T^{2} \)
13 \( 1 - 1.14e6T + 1.79e12T^{2} \)
17 \( 1 + 3.04e6iT - 3.42e13T^{2} \)
19 \( 1 - 4.99e6iT - 1.16e14T^{2} \)
23 \( 1 + 5.14e6T + 9.52e14T^{2} \)
29 \( 1 - 1.44e8iT - 1.22e16T^{2} \)
31 \( 1 + 2.51e7iT - 2.54e16T^{2} \)
37 \( 1 + 1.69e8T + 1.77e17T^{2} \)
41 \( 1 + 3.25e8iT - 5.50e17T^{2} \)
43 \( 1 + 1.17e9iT - 9.29e17T^{2} \)
47 \( 1 - 1.82e9T + 2.47e18T^{2} \)
53 \( 1 - 5.85e7iT - 9.26e18T^{2} \)
59 \( 1 - 5.59e8T + 3.01e19T^{2} \)
61 \( 1 - 8.08e9T + 4.35e19T^{2} \)
67 \( 1 - 2.71e9iT - 1.22e20T^{2} \)
71 \( 1 - 1.89e10T + 2.31e20T^{2} \)
73 \( 1 - 1.88e10T + 3.13e20T^{2} \)
79 \( 1 - 4.26e10iT - 7.47e20T^{2} \)
83 \( 1 - 3.76e10T + 1.28e21T^{2} \)
89 \( 1 + 5.79e10iT - 2.77e21T^{2} \)
97 \( 1 + 9.56e10T + 7.15e21T^{2} \)
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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

−16.69082465658402649144660869551, −15.73194408826281824463097391467, −13.96106237683877808101032062274, −12.73912463905332679377820141395, −10.95246902234489243544690005871, −10.31116946991292766360821265912, −7.01185110093270582188829957111, −5.18976620496976722812675348382, −3.74594787929686565732794857212, −0.792958304431124367202459349773, 2.42446969345341535011621234462, 5.18334883947060185119635115392, 6.23416158813341109875066658625, 8.165932564463743469937350176172, 11.03892302686960963705305128453, 12.30274000859341654475036616101, 13.27101610396735912237019915585, 15.25944960554504540524378931502, 16.11540537352178631089530715835, 17.82261609909780795712758999032

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