Properties

Label 2-930-31.8-c1-0-2
Degree $2$
Conductor $930$
Sign $-0.226 - 0.973i$
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 + (−2.70 − 1.96i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (2.20 + 1.60i)11-s + (0.309 − 0.951i)12-s + (−0.0554 − 0.170i)13-s + (−2.70 + 1.96i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.444 + 0.322i)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 + (−1.02 − 0.743i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (0.665 + 0.483i)11-s + (0.0892 − 0.274i)12-s + (−0.0153 − 0.0473i)13-s + (−0.723 + 0.525i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.107 + 0.0783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.226 - 0.973i$
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: \((2,\ 930,\ (\ :1/2),\ -0.226 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362328 + 0.456283i\)
\(L(\frac12)\) \(\approx\) \(0.362328 + 0.456283i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.685 - 5.52i)T \)
good7 \( 1 + (2.70 + 1.96i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-2.20 - 1.60i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0554 + 0.170i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.444 - 0.322i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.78 - 5.50i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.77 - 4.19i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.614 - 1.89i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 4.34T + 37T^{2} \)
41 \( 1 + (1.99 - 6.14i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (0.575 - 1.77i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.46 + 4.51i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.17 + 2.30i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.199 + 0.613i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 8.75T + 67T^{2} \)
71 \( 1 + (-8.57 + 6.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-7.43 - 5.40i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.66 - 4.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.98 + 6.12i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.842 - 0.612i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.54 - 1.85i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.16492078601683683843321516454, −9.803993136642168001125758957543, −8.864053350633308391565663641261, −7.896299849603792046955337310181, −6.85603705069825725866663355780, −5.93066853642345365338863614399, −4.68042790683302880026003520993, −3.77005614328862583652179704174, −3.34049811083734655774143634016, −1.68379059480303897281956784847, 0.24237805806458248703482017090, 2.39291729415706435644086157856, 3.46398622548532975211171586331, 4.48237947394565940701759098788, 5.82800913255062977389242164098, 6.39283086264760599435936164224, 7.13552796170697067820317908579, 8.127537486676184993664617056523, 8.906422323084128174556112843397, 9.427328122146751405170618362543

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