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Properties

Label 2-930-155.4-c1-0-17
Degree $2$
Conductor $930$
Sign $0.950 + 0.312i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  + (0.951 − 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (2.21 + 0.332i)5-s − 0.999·6-s + (0.907 + 1.24i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (2.20 − 0.367i)10-s + (1.74 − 1.27i)11-s + (−0.951 + 0.309i)12-s + (−0.363 − 0.118i)13-s + (1.24 + 0.907i)14-s + (−2.00 − 0.999i)15-s + (0.309 − 0.951i)16-s + (0.638 − 0.879i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.549 − 0.178i)3-s + (0.404 − 0.293i)4-s + (0.988 + 0.148i)5-s − 0.408·6-s + (0.342 + 0.471i)7-s + (0.207 − 0.286i)8-s + (0.269 + 0.195i)9-s + (0.697 − 0.116i)10-s + (0.527 − 0.383i)11-s + (−0.274 + 0.0892i)12-s + (−0.100 − 0.0327i)13-s + (0.333 + 0.242i)14-s + (−0.516 − 0.257i)15-s + (0.0772 − 0.237i)16-s + (0.154 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.950 + 0.312i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.950 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50592 - 0.400953i\)
\(L(\frac12)\) \(\approx\) \(2.50592 - 0.400953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-2.21 - 0.332i)T \)
31 \( 1 + (-5.47 + 1.01i)T \)
good7 \( 1 + (-0.907 - 1.24i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-1.74 + 1.27i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.363 + 0.118i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.638 + 0.879i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.32 - 4.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.62 - 3.61i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.31 + 4.05i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 3.70iT - 37T^{2} \)
41 \( 1 + (-0.587 - 1.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (7.37 - 2.39i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-9.01 - 2.92i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.47 + 4.78i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.08 + 3.33i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 4.97T + 61T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (4.31 + 3.13i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.86 + 5.32i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.01 - 2.91i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.57 - 0.512i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.341 - 0.248i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (8.18 + 11.2i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.07472281640626371314771948448, −9.487382177740762111522573526095, −8.321050868685507842928681243361, −7.25963682945158242292567742603, −6.18470573386407612889714863049, −5.77576724158036729166862575973, −4.92283055478896282372035335301, −3.71326233645429822458142199230, −2.42614424017791651580609030454, −1.38036237328263060707813157390, 1.33833282134612237911271594354, 2.68861456971514494358464000001, 4.13193886585312078847274856357, 4.86676466907403245188980855524, 5.69493955378992300286334059139, 6.58928667242422207965262190536, 7.19726491403596008924897337352, 8.457191705251392019988453966136, 9.375302281079798715265468898716, 10.29040549640905476297732225778

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