Properties

Label 2-354-177.176-c5-0-60
Degree $2$
Conductor $354$
Sign $-0.562 + 0.826i$
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 + (9.34 − 12.4i)3-s + 16·4-s + 88.1i·5-s + (−37.3 + 49.9i)6-s − 156.·7-s − 64·8-s + (−68.4 − 233. i)9-s − 352. i·10-s + 58.7·11-s + (149. − 199. i)12-s + 776. i·13-s + 627.·14-s + (1.10e3 + 823. i)15-s + 256·16-s − 132. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.599 − 0.800i)3-s + 0.5·4-s + 1.57i·5-s + (−0.423 + 0.566i)6-s − 1.21·7-s − 0.353·8-s + (−0.281 − 0.959i)9-s − 1.11i·10-s + 0.146·11-s + (0.299 − 0.400i)12-s + 1.27i·13-s + 0.856·14-s + (1.26 + 0.945i)15-s + 0.250·16-s − 0.110i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.562 + 0.826i$
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: \((2,\ 354,\ (\ :5/2),\ -0.562 + 0.826i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4382284592\)
\(L(\frac12)\) \(\approx\) \(0.4382284592\)
\(L(\frac{7}{2})\) 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 \( 1 + (-9.34 + 12.4i)T \)
59 \( 1 + (-2.67e4 + 1.21e3i)T \)
good5 \( 1 - 88.1iT - 3.12e3T^{2} \)
7 \( 1 + 156.T + 1.68e4T^{2} \)
11 \( 1 - 58.7T + 1.61e5T^{2} \)
13 \( 1 - 776. iT - 3.71e5T^{2} \)
17 \( 1 + 132. iT - 1.41e6T^{2} \)
19 \( 1 + 134.T + 2.47e6T^{2} \)
23 \( 1 - 566.T + 6.43e6T^{2} \)
29 \( 1 + 4.41e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.18e3iT - 2.86e7T^{2} \)
37 \( 1 + 3.15e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.55e3iT - 1.15e8T^{2} \)
43 \( 1 + 933. iT - 1.47e8T^{2} \)
47 \( 1 + 6.09e3T + 2.29e8T^{2} \)
53 \( 1 + 3.47e4iT - 4.18e8T^{2} \)
61 \( 1 + 3.79e3iT - 8.44e8T^{2} \)
67 \( 1 + 2.54e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.34e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.94e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.33e4T + 3.07e9T^{2} \)
83 \( 1 + 3.49e4T + 3.93e9T^{2} \)
89 \( 1 - 4.98e4T + 5.58e9T^{2} \)
97 \( 1 + 9.78e4iT - 8.58e9T^{2} \)
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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

−10.11010783834357310209424763812, −9.445836858693306370403131163992, −8.480461314378944762716903148743, −7.17361354009114444207772651523, −6.83047280371812419508258771951, −6.13543954071182099552378352788, −3.67343931722980913230970530673, −2.85978715694472746516341619923, −1.88154240161267300115254897641, −0.14256441825431767394440743661, 1.05151009547192550047002926037, 2.70369854107096992252261396340, 3.82164728542049416612314955129, 5.05140184784904504923525211382, 6.02918474105904208030685131123, 7.60573422932811061195977132937, 8.457079825777760358192460913072, 9.152750491163030531109164242348, 9.788585661179226169377290702694, 10.54918547621539873599795465107

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