Properties

Label 2-930-31.9-c1-0-10
Degree $2$
Conductor $930$
Sign $0.975 - 0.218i$
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 + (−0.978 − 0.207i)3-s + (−0.809 − 0.587i)4-s + (0.5 + 0.866i)5-s + (0.5 − 0.866i)6-s + (3.87 − 1.72i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.300 + 2.85i)11-s + (0.669 + 0.743i)12-s + (2.45 − 2.72i)13-s + (0.442 + 4.21i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.133 − 1.27i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.564 − 0.120i)3-s + (−0.404 − 0.293i)4-s + (0.223 + 0.387i)5-s + (0.204 − 0.353i)6-s + (1.46 − 0.651i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (−0.309 + 0.0657i)10-s + (−0.0904 + 0.860i)11-s + (0.193 + 0.214i)12-s + (0.680 − 0.755i)13-s + (0.118 + 1.12i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.0324 − 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.975 - 0.218i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.975 - 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35894 + 0.150624i\)
\(L(\frac12)\) \(\approx\) \(1.35894 + 0.150624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-5.42 + 1.23i)T \)
good7 \( 1 + (-3.87 + 1.72i)T + (4.68 - 5.20i)T^{2} \)
11 \( 1 + (0.300 - 2.85i)T + (-10.7 - 2.28i)T^{2} \)
13 \( 1 + (-2.45 + 2.72i)T + (-1.35 - 12.9i)T^{2} \)
17 \( 1 + (0.133 + 1.27i)T + (-16.6 + 3.53i)T^{2} \)
19 \( 1 + (5.60 + 6.22i)T + (-1.98 + 18.8i)T^{2} \)
23 \( 1 + (0.267 - 0.194i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.60 + 4.94i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (-0.905 + 1.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.07 + 1.50i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-6.91 - 7.68i)T + (-4.49 + 42.7i)T^{2} \)
47 \( 1 + (-2.82 - 8.70i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.23 + 4.10i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (0.142 + 0.0302i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 - 4.05T + 61T^{2} \)
67 \( 1 + (0.978 + 1.69i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.96 - 3.10i)T + (47.5 + 52.7i)T^{2} \)
73 \( 1 + (0.627 - 5.96i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (1.10 + 10.4i)T + (-77.2 + 16.4i)T^{2} \)
83 \( 1 + (0.858 - 0.182i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (-7.85 - 5.70i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-10.7 - 7.82i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.19437255713287817056304700565, −9.219422517617480417995806226836, −8.095795271795025131977689204140, −7.62469259154739201821479227833, −6.69661204497375219981649136774, −5.91869154178530091105169920001, −4.76072295187391046216351829438, −4.33757081840226425683601897536, −2.36395890832106140792060277825, −0.920081683926832819337262153608, 1.24495951018514744837589507201, 2.17268225171612407652607311648, 3.82526855382872596857514738238, 4.66547235598220363203432340796, 5.60648723142989816342206456304, 6.35430785870364045016243002515, 7.897007506844845889133749845210, 8.544500156316896055714254778103, 9.054403878659586499647540499801, 10.36594113142116003747862041499

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