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Properties

Label 2-930-31.8-c1-0-23
Degree $2$
Conductor $930$
Sign $-0.424 - 0.905i$
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  + (0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 5-s − 0.999·6-s + (0.775 + 0.563i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (−2.21 − 1.60i)11-s + (−0.309 + 0.951i)12-s + (−1.28 − 3.95i)13-s + (0.775 − 0.563i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−1.90 + 1.38i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.178 − 0.549i)3-s + (−0.404 − 0.293i)4-s − 0.447·5-s − 0.408·6-s + (0.293 + 0.212i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (−0.667 − 0.484i)11-s + (−0.0892 + 0.274i)12-s + (−0.356 − 1.09i)13-s + (0.207 − 0.150i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.461 + 0.334i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113411 + 0.178403i\)
\(L(\frac12)\) \(\approx\) \(0.113411 + 0.178403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + T \)
31 \( 1 + (0.480 + 5.54i)T \)
good7 \( 1 + (-0.775 - 0.563i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (2.21 + 1.60i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.28 + 3.95i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.90 - 1.38i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.657 - 2.02i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.05 - 3.67i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.74 - 5.37i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 6.85T + 37T^{2} \)
41 \( 1 + (1.07 - 3.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.42 - 7.47i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.41 + 7.42i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.50 - 1.09i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.11 - 6.49i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 8.57T + 61T^{2} \)
67 \( 1 - 8.28T + 67T^{2} \)
71 \( 1 + (-2.58 + 1.87i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.40 + 1.02i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.56 - 3.32i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.73 - 8.42i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.57 + 5.50i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (12.9 + 9.38i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.731803549417845430577364147533, −8.379545191682215592602607362971, −8.087313732177722528824028005900, −6.98538551309443372571561820410, −5.75336368050423441907698713930, −5.19558645773010551061725062204, −3.89508056394697183975989779452, −2.89903379491080365707315434490, −1.71853039814928579652663931482, −0.090111790089249815258317624635, 2.29737570336328270543810574599, 3.79646872613252975058932467161, 4.57426682468502316446026671530, 5.21759129263837941312383434145, 6.48001582106132052520662209487, 7.15684307932262804004874153862, 8.086854093757803351310714474364, 8.867209888380940523221616650606, 9.747373129319741485807541578225, 10.56514148181345255341858844607

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