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Properties

Label 2-354-177.176-c5-0-13
Degree $2$
Conductor $354$
Sign $0.710 - 0.703i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + (−12.9 − 8.73i)3-s + 16·4-s − 27.1i·5-s + (−51.6 − 34.9i)6-s − 249.·7-s + 64·8-s + (90.3 + 225. i)9-s − 108. i·10-s + 19.0·11-s + (−206. − 139. i)12-s − 831. i·13-s − 997.·14-s + (−237. + 350. i)15-s + 256·16-s − 32.0i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.828 − 0.560i)3-s + 0.5·4-s − 0.486i·5-s + (−0.585 − 0.396i)6-s − 1.92·7-s + 0.353·8-s + (0.371 + 0.928i)9-s − 0.343i·10-s + 0.0475·11-s + (−0.414 − 0.280i)12-s − 1.36i·13-s − 1.36·14-s + (−0.272 + 0.402i)15-s + 0.250·16-s − 0.0269i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8652264703\)
\(L(\frac12)\) \(\approx\) \(0.8652264703\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4T \)
3 \( 1 + (12.9 + 8.73i)T \)
59 \( 1 + (5.19e3 - 2.62e4i)T \)
good5 \( 1 + 27.1iT - 3.12e3T^{2} \)
7 \( 1 + 249.T + 1.68e4T^{2} \)
11 \( 1 - 19.0T + 1.61e5T^{2} \)
13 \( 1 + 831. iT - 3.71e5T^{2} \)
17 \( 1 + 32.0iT - 1.41e6T^{2} \)
19 \( 1 + 1.97e3T + 2.47e6T^{2} \)
23 \( 1 + 2.85e3T + 6.43e6T^{2} \)
29 \( 1 - 5.58e3iT - 2.05e7T^{2} \)
31 \( 1 + 735. iT - 2.86e7T^{2} \)
37 \( 1 - 1.26e4iT - 6.93e7T^{2} \)
41 \( 1 + 553. iT - 1.15e8T^{2} \)
43 \( 1 + 1.31e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.58e4T + 2.29e8T^{2} \)
53 \( 1 - 3.17e4iT - 4.18e8T^{2} \)
61 \( 1 + 4.86e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.14e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.85e4iT - 1.80e9T^{2} \)
73 \( 1 - 5.75e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.57e4T + 3.07e9T^{2} \)
83 \( 1 - 1.17e5T + 3.93e9T^{2} \)
89 \( 1 + 3.77e4T + 5.58e9T^{2} \)
97 \( 1 - 6.47e4iT - 8.58e9T^{2} \)
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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

−10.64374950936441289069193902138, −10.26087552628476423855268443725, −8.893024553726718819887117476012, −7.62407792823618502648288516510, −6.55189778098433272755395435444, −6.02724454657230438847102815973, −5.02787409050284283593571830906, −3.69859509285757420973276702903, −2.52583771949465276013303196727, −0.826237114814405317358182303911, 0.25122184161891104510554008604, 2.38171883034482310320802786812, 3.70134595040536780076965840628, 4.31577548128644827722372271505, 5.90936503565300756231686361405, 6.40279353756565806290319467255, 7.09666826779776500917282349144, 8.999957088552504373564603490237, 9.853787213720754045248975263066, 10.55350597171126910207325339511

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