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Properties

Label 2-930-465.464-c1-0-48
Degree $2$
Conductor $930$
Sign $0.522 + 0.852i$
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  − 2-s + (1.63 + 0.571i)3-s + 4-s + (1.46 − 1.69i)5-s + (−1.63 − 0.571i)6-s − 2.83i·7-s − 8-s + (2.34 + 1.86i)9-s + (−1.46 + 1.69i)10-s − 3.85·11-s + (1.63 + 0.571i)12-s − 2.42·13-s + 2.83i·14-s + (3.35 − 1.92i)15-s + 16-s − 4.97i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.943 + 0.330i)3-s + 0.5·4-s + (0.654 − 0.756i)5-s + (−0.667 − 0.233i)6-s − 1.07i·7-s − 0.353·8-s + (0.782 + 0.623i)9-s + (−0.462 + 0.534i)10-s − 1.16·11-s + (0.471 + 0.165i)12-s − 0.672·13-s + 0.758i·14-s + (0.867 − 0.497i)15-s + 0.250·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44134 - 0.807708i\)
\(L(\frac12)\) \(\approx\) \(1.44134 - 0.807708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.63 - 0.571i)T \)
5 \( 1 + (-1.46 + 1.69i)T \)
31 \( 1 + (0.156 + 5.56i)T \)
good7 \( 1 + 2.83iT - 7T^{2} \)
11 \( 1 + 3.85T + 11T^{2} \)
13 \( 1 + 2.42T + 13T^{2} \)
17 \( 1 + 4.97iT - 17T^{2} \)
19 \( 1 - 8.42T + 19T^{2} \)
23 \( 1 - 0.144iT - 23T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
37 \( 1 + 8.07T + 37T^{2} \)
41 \( 1 - 7.56iT - 41T^{2} \)
43 \( 1 - 0.244T + 43T^{2} \)
47 \( 1 - 0.441T + 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 10.1iT - 61T^{2} \)
67 \( 1 + 1.92iT - 67T^{2} \)
71 \( 1 + 0.737iT - 71T^{2} \)
73 \( 1 + 4.18T + 73T^{2} \)
79 \( 1 - 14.3iT - 79T^{2} \)
83 \( 1 - 4.53iT - 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 11.9iT - 97T^{2} \)
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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

−9.869505060521850889700885347368, −9.297468158960126805055413963412, −8.238277412051813910396793089279, −7.61782918743080136017611083834, −6.96116903407098405123068882421, −5.33219340206813645458447646133, −4.72215314885523958098833455690, −3.28213687073518546019184829403, −2.31017108134997050574149064568, −0.893757209759963499314376287322, 1.70541066511282455132798365259, 2.63657397425738694221096209184, 3.27035890759803854030166612299, 5.19418698190966397609011782169, 6.03509809615498434836789435521, 7.13324583870156669848790116314, 7.68502808048675626208633492648, 8.665950836553006918861005628278, 9.236859499693032992540178675601, 10.19416643572720306347307403728

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