Properties

Label 2-12-12.11-c17-0-25
Degree $2$
Conductor $12$
Sign $0.263 + 0.964i$
Analytic cond. $21.9866$
Root an. cond. $4.68899$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−207. + 296. i)2-s + (1.13e4 + 959. i)3-s + (−4.51e4 − 1.23e5i)4-s + 8.04e5i·5-s + (−2.63e6 + 3.16e6i)6-s − 2.45e7i·7-s + (4.58e7 + 1.21e7i)8-s + (1.27e8 + 2.17e7i)9-s + (−2.38e8 − 1.66e8i)10-s − 7.53e8·11-s + (−3.92e8 − 1.43e9i)12-s − 5.41e9·13-s + (7.27e9 + 5.08e9i)14-s + (−7.71e8 + 9.11e9i)15-s + (−1.31e10 + 1.11e10i)16-s − 2.92e10i·17-s + ⋯
L(s)  = 1  + (−0.572 + 0.819i)2-s + (0.996 + 0.0844i)3-s + (−0.344 − 0.938i)4-s + 0.921i·5-s + (−0.639 + 0.768i)6-s − 1.60i·7-s + (0.966 + 0.255i)8-s + (0.985 + 0.168i)9-s + (−0.755 − 0.527i)10-s − 1.05·11-s + (−0.263 − 0.964i)12-s − 1.84·13-s + (1.31 + 0.920i)14-s + (−0.0777 + 0.917i)15-s + (−0.763 + 0.646i)16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12\)    =    \(2^{2} \cdot 3\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(21.9866\)
Root analytic conductor: \(4.68899\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{12} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 12,\ (\ :17/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.738437 - 0.563690i\)
\(L(\frac12)\) \(\approx\) \(0.738437 - 0.563690i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (207. - 296. i)T \)
3 \( 1 + (-1.13e4 - 959. i)T \)
good5 \( 1 - 8.04e5iT - 7.62e11T^{2} \)
7 \( 1 + 2.45e7iT - 2.32e14T^{2} \)
11 \( 1 + 7.53e8T + 5.05e17T^{2} \)
13 \( 1 + 5.41e9T + 8.65e18T^{2} \)
17 \( 1 + 2.92e10iT - 8.27e20T^{2} \)
19 \( 1 + 4.12e10iT - 5.48e21T^{2} \)
23 \( 1 + 2.06e11T + 1.41e23T^{2} \)
29 \( 1 + 2.39e12iT - 7.25e24T^{2} \)
31 \( 1 + 2.92e12iT - 2.25e25T^{2} \)
37 \( 1 - 8.68e12T + 4.56e26T^{2} \)
41 \( 1 + 1.59e13iT - 2.61e27T^{2} \)
43 \( 1 + 7.46e13iT - 5.87e27T^{2} \)
47 \( 1 + 5.30e12T + 2.66e28T^{2} \)
53 \( 1 - 5.59e14iT - 2.05e29T^{2} \)
59 \( 1 + 8.13e13T + 1.27e30T^{2} \)
61 \( 1 + 1.52e15T + 2.24e30T^{2} \)
67 \( 1 + 3.40e14iT - 1.10e31T^{2} \)
71 \( 1 + 2.49e15T + 2.96e31T^{2} \)
73 \( 1 + 3.35e15T + 4.74e31T^{2} \)
79 \( 1 - 7.33e15iT - 1.81e32T^{2} \)
83 \( 1 + 2.82e14T + 4.21e32T^{2} \)
89 \( 1 + 9.20e15iT - 1.37e33T^{2} \)
97 \( 1 + 3.56e16T + 5.95e33T^{2} \)
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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

−15.49523836212097302955925965396, −14.37242794752229175410484458705, −13.53032700406041691949485810309, −10.53975816799053002371657766722, −9.720440515136969061754289095189, −7.61775112264426406569506946465, −7.16801730216514992440539656645, −4.57573502789876270769352464859, −2.54367995670929983182923849677, −0.32687327086896021994377035777, 1.80163393293498469310700594255, 2.83107151710174500849074849284, 4.89212409993510084733258125584, 7.910401982956098334672993037148, 8.850184542087808164412250382946, 9.982151429755327455178480972700, 12.30440515146183005903475469308, 12.79823337797593560543796736765, 14.80863164970376099182854037432, 16.25870590981928439309925223147

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