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Properties

Label 2-930-31.4-c1-0-19
Degree $2$
Conductor $930$
Sign $0.549 + 0.835i$
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 + (2.40 − 1.74i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−2.90 + 2.11i)11-s + (0.309 + 0.951i)12-s + (1.64 − 5.04i)13-s + (2.40 + 1.74i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−2.14 − 1.55i)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 + (0.910 − 0.661i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (−0.876 + 0.637i)11-s + (0.0892 + 0.274i)12-s + (0.454 − 1.39i)13-s + (0.643 + 0.467i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.519 − 0.377i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.549 + 0.835i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.549 + 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29283 - 0.696877i\)
\(L(\frac12)\) \(\approx\) \(1.29283 - 0.696877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.47 + 3.31i)T \)
good7 \( 1 + (-2.40 + 1.74i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.90 - 2.11i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.64 + 5.04i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.14 + 1.55i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.52 + 4.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.61 - 5.53i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.95 + 9.10i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 3.78T + 37T^{2} \)
41 \( 1 + (-0.660 - 2.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.52 + 4.69i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.29 + 3.97i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.45 + 1.05i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.98 + 9.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 5.28T + 61T^{2} \)
67 \( 1 + 2.75T + 67T^{2} \)
71 \( 1 + (-6.04 - 4.39i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.27 + 1.65i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-11.5 - 8.38i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.97 + 6.09i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (1.11 - 0.807i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (9.32 - 6.77i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.830783282677842582972993614146, −8.806531903940828530106369610336, −7.82465763036020526653262849089, −7.62053458758130563321460841678, −6.75603623915071452905529494112, −5.46851276140147306926082078390, −4.82464393464228825060865039322, −3.69564109685940132367823584400, −2.43408819854790916584842249810, −0.64842271840456376699459373171, 1.64630047862879363692283775889, 2.82152371080061491102832297244, 3.90047227518820182350240467068, 4.76881215231203700657975338097, 5.51634077203303410982476638380, 6.71543145179464627664275680836, 8.029449893639021839215273840731, 8.760624980479143775224647327457, 9.162455050613152294233870425896, 10.64142323496812649336767527112

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