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Properties

Label 2-351-117.4-c1-0-11
Degree $2$
Conductor $351$
Sign $0.362 - 0.932i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.65i·2-s − 5.03·4-s + (−2.43 − 1.40i)5-s + (0.226 + 0.130i)7-s + 8.03i·8-s + (−3.72 + 6.44i)10-s + 1.55i·11-s + (−2.26 + 2.80i)13-s + (0.346 − 0.599i)14-s + 11.2·16-s + (−3.65 − 6.32i)17-s + (0.447 − 0.258i)19-s + (12.2 + 7.06i)20-s + 4.13·22-s + (1.37 + 2.38i)23-s + ⋯
L(s)  = 1  − 1.87i·2-s − 2.51·4-s + (−1.08 − 0.627i)5-s + (0.0854 + 0.0493i)7-s + 2.84i·8-s + (−1.17 + 2.03i)10-s + 0.469i·11-s + (−0.627 + 0.778i)13-s + (0.0925 − 0.160i)14-s + 2.81·16-s + (−0.886 − 1.53i)17-s + (0.102 − 0.0592i)19-s + (2.73 + 1.57i)20-s + 0.881·22-s + (0.286 + 0.496i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.362 - 0.932i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.362 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198459 + 0.135819i\)
\(L(\frac12)\) \(\approx\) \(0.198459 + 0.135819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (2.26 - 2.80i)T \)
good2 \( 1 + 2.65iT - 2T^{2} \)
5 \( 1 + (2.43 + 1.40i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.226 - 0.130i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
17 \( 1 + (3.65 + 6.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.447 + 0.258i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.37 - 2.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.56T + 29T^{2} \)
31 \( 1 + (3.17 + 1.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 1.10i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.42 - 3.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.74 + 3.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.10 - 4.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.68T + 53T^{2} \)
59 \( 1 + 11.1iT - 59T^{2} \)
61 \( 1 + (-1.80 + 3.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.11 - 4.10i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.40 + 3.12i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.76iT - 73T^{2} \)
79 \( 1 + (1.36 + 2.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.18 - 4.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.18 - 0.684i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.7 + 6.77i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.28030312773011358781692652993, −9.771828053044749893629917137050, −9.290111415717788177016689574231, −8.311939437804222925616070691123, −7.18326137574660705811837422870, −4.96291830805655502866465356179, −4.46791407182896903485429909990, −3.30199448245719819911726195926, −1.93604241604302730979587787708, −0.16608516262183838586976192240, 3.49582820694565403180079071331, 4.50170239477374976714202153811, 5.68821080815505927972175343121, 6.64642740379681000164339750226, 7.48593731881105395348467343717, 8.160464233648678371513559308012, 8.922711414871770262903420644614, 10.24890794806078736757317434669, 11.19300986945731961276972686141, 12.56395121620933486815650578907

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