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Properties

Label 2-930-93.92-c1-0-14
Degree $2$
Conductor $930$
Sign $0.823 - 0.567i$
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  + i·2-s + (−1.58 + 0.687i)3-s − 4-s + i·5-s + (−0.687 − 1.58i)6-s − 4.71·7-s − i·8-s + (2.05 − 2.18i)9-s − 10-s + 4.26·11-s + (1.58 − 0.687i)12-s − 5.10i·13-s − 4.71i·14-s + (−0.687 − 1.58i)15-s + 16-s − 2.21·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.917 + 0.396i)3-s − 0.5·4-s + 0.447i·5-s + (−0.280 − 0.649i)6-s − 1.78·7-s − 0.353i·8-s + (0.685 − 0.728i)9-s − 0.316·10-s + 1.28·11-s + (0.458 − 0.198i)12-s − 1.41i·13-s − 1.26i·14-s + (−0.177 − 0.410i)15-s + 0.250·16-s − 0.537·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.823 - 0.567i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.823 - 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.747016 + 0.232423i\)
\(L(\frac12)\) \(\approx\) \(0.747016 + 0.232423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.58 - 0.687i)T \)
5 \( 1 - iT \)
31 \( 1 + (5.46 - 1.08i)T \)
good7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
17 \( 1 + 2.21T + 17T^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 - 2.88T + 23T^{2} \)
29 \( 1 - 9.48T + 29T^{2} \)
37 \( 1 + 1.37iT - 37T^{2} \)
41 \( 1 - 9.55iT - 41T^{2} \)
43 \( 1 - 2.98iT - 43T^{2} \)
47 \( 1 + 5.85iT - 47T^{2} \)
53 \( 1 - 8.63T + 53T^{2} \)
59 \( 1 + 4.95iT - 59T^{2} \)
61 \( 1 + 7.20iT - 61T^{2} \)
67 \( 1 - 16.2T + 67T^{2} \)
71 \( 1 - 6.04iT - 71T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 + 3.18iT - 79T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 - 5.58T + 89T^{2} \)
97 \( 1 - 2.28T + 97T^{2} \)
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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

−10.03754947734496269410025654134, −9.476394006327822135936807244875, −8.581979899653123902459707703348, −7.18903034760859569733275892338, −6.49489244249357992738008481456, −6.17119335471304977158929256391, −5.07661042911237307183138244423, −3.88579574389077092691170031376, −3.11978112055737316724811434605, −0.59816173898315033292937034936, 0.890133838587161449379217408865, 2.25460051376041230869463115610, 3.76663512217485813759367903186, 4.45620288804919018090424677578, 5.73958038545247520749621021478, 6.66639808394377286466703654729, 6.97675489506955851025288965955, 8.743627999741551388663633655651, 9.253979763838322999484320940240, 10.02200334225058723197289634207

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