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Properties

Label 2-12-12.11-c13-0-15
Degree $2$
Conductor $12$
Sign $0.985 - 0.167i$
Analytic cond. $12.8677$
Root an. cond. $3.58715$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−26.1 + 86.6i)2-s + (919. − 865. i)3-s + (−6.82e3 − 4.53e3i)4-s + 3.47e4i·5-s + (5.08e4 + 1.02e5i)6-s − 2.98e5i·7-s + (5.71e5 − 4.72e5i)8-s + (9.67e4 − 1.59e6i)9-s + (−3.01e6 − 9.11e5i)10-s + 2.90e6·11-s + (−1.01e7 + 1.72e6i)12-s + 2.73e7·13-s + (2.58e7 + 7.80e6i)14-s + (3.01e7 + 3.19e7i)15-s + (2.59e7 + 6.19e7i)16-s − 2.14e7i·17-s + ⋯
L(s)  = 1  + (−0.289 + 0.957i)2-s + (0.728 − 0.685i)3-s + (−0.832 − 0.553i)4-s + 0.995i·5-s + (0.445 + 0.895i)6-s − 0.957i·7-s + (0.771 − 0.636i)8-s + (0.0607 − 0.998i)9-s + (−0.953 − 0.288i)10-s + 0.494·11-s + (−0.985 + 0.167i)12-s + 1.57·13-s + (0.916 + 0.277i)14-s + (0.682 + 0.725i)15-s + (0.386 + 0.922i)16-s − 0.215i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(12.8677\)
Root analytic conductor: \(3.58715\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :13/2),\ 0.985 - 0.167i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.93703 + 0.162993i\)
\(L(\frac12)\) \(\approx\) \(1.93703 + 0.162993i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (26.1 - 86.6i)T \)
3 \( 1 + (-919. + 865. i)T \)
good5 \( 1 - 3.47e4iT - 1.22e9T^{2} \)
7 \( 1 + 2.98e5iT - 9.68e10T^{2} \)
11 \( 1 - 2.90e6T + 3.45e13T^{2} \)
13 \( 1 - 2.73e7T + 3.02e14T^{2} \)
17 \( 1 + 2.14e7iT - 9.90e15T^{2} \)
19 \( 1 - 2.70e7iT - 4.20e16T^{2} \)
23 \( 1 - 1.21e9T + 5.04e17T^{2} \)
29 \( 1 - 1.45e9iT - 1.02e19T^{2} \)
31 \( 1 + 8.52e9iT - 2.44e19T^{2} \)
37 \( 1 + 2.62e10T + 2.43e20T^{2} \)
41 \( 1 + 2.12e10iT - 9.25e20T^{2} \)
43 \( 1 - 2.09e10iT - 1.71e21T^{2} \)
47 \( 1 - 2.83e10T + 5.46e21T^{2} \)
53 \( 1 - 1.84e10iT - 2.60e22T^{2} \)
59 \( 1 + 4.65e11T + 1.04e23T^{2} \)
61 \( 1 + 4.08e11T + 1.61e23T^{2} \)
67 \( 1 - 1.14e12iT - 5.48e23T^{2} \)
71 \( 1 - 9.35e11T + 1.16e24T^{2} \)
73 \( 1 - 5.16e11T + 1.67e24T^{2} \)
79 \( 1 + 2.03e12iT - 4.66e24T^{2} \)
83 \( 1 - 1.79e12T + 8.87e24T^{2} \)
89 \( 1 - 6.29e12iT - 2.19e25T^{2} \)
97 \( 1 - 3.83e12T + 6.73e25T^{2} \)
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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

−17.06236035356355110144452807696, −15.33710470369740281508361692600, −14.19148786006138790327332070938, −13.33254385751990310131912036655, −10.76569526964745027376667976388, −8.966876961818136385666631096990, −7.41281348731672174181043970447, −6.45603443408383411883485930396, −3.63504833867889516060688278345, −1.05729364339875519224012026360, 1.44697462800593740055596991428, 3.32344916236647458540934074185, 4.95413457019633240626612748495, 8.624161825260076564836209598310, 9.041265418335731906325509183130, 10.86679964846379827231172325910, 12.47959138509138148382885967847, 13.75915785439531301306346108235, 15.55186764993689318444960704126, 16.89553740502218199773007679900

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