Properties

Label 2-930-31.7-c1-0-15
Degree $2$
Conductor $930$
Sign $-0.999 + 0.0184i$
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 + (−0.381 − 0.169i)7-s + (0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (−0.275 − 2.62i)11-s + (0.669 − 0.743i)12-s + (−1.91 − 2.13i)13-s + (−0.0436 + 0.415i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.588 + 5.59i)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 + (−0.144 − 0.0641i)7-s + (0.286 + 0.207i)8-s + (0.304 − 0.135i)9-s + (−0.309 − 0.0657i)10-s + (−0.0832 − 0.791i)11-s + (0.193 − 0.214i)12-s + (−0.532 − 0.590i)13-s + (−0.0116 + 0.110i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.142 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.999 + 0.0184i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.999 + 0.0184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00497169 - 0.539312i\)
\(L(\frac12)\) \(\approx\) \(0.00497169 - 0.539312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.56 - 0.0902i)T \)
good7 \( 1 + (0.381 + 0.169i)T + (4.68 + 5.20i)T^{2} \)
11 \( 1 + (0.275 + 2.62i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (1.91 + 2.13i)T + (-1.35 + 12.9i)T^{2} \)
17 \( 1 + (0.588 - 5.59i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-1.97 + 2.19i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-2.85 - 2.07i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.59 + 7.97i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (3.43 + 5.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.6 + 2.27i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (-6.56 + 7.29i)T + (-4.49 - 42.7i)T^{2} \)
47 \( 1 + (3.01 - 9.27i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.06 - 1.36i)T + (35.4 - 39.3i)T^{2} \)
59 \( 1 + (1.66 - 0.354i)T + (53.8 - 23.9i)T^{2} \)
61 \( 1 + 9.40T + 61T^{2} \)
67 \( 1 + (0.288 - 0.499i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.53 - 1.12i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (-0.450 - 4.29i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (0.594 - 5.65i)T + (-77.2 - 16.4i)T^{2} \)
83 \( 1 + (9.39 + 1.99i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 + (-3.08 + 2.24i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.33 + 4.60i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.751646517418998135857273704650, −9.008007670925050000058438888985, −8.135355384861944689208991560607, −7.18698743956641144258128696509, −5.93104999350488249520469286563, −5.31192150529869796771042882360, −4.16420378991550734586625062691, −3.17721357845145487398729153812, −1.75271053036983652171537243322, −0.29246576216084647595337838560, 1.67533263761856253298673862321, 3.18221444200056560098976089047, 4.73244168004836060877838958164, 5.21891659227444139268975604553, 6.42853488925509597254429198140, 7.04763373897734168814833685417, 7.62027917362900509089199015141, 8.942098649100682870447928945242, 9.602175361273664324559155302377, 10.31297521513584344001153236908

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