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Properties

Label 2-930-155.4-c1-0-6
Degree $2$
Conductor $930$
Sign $0.149 - 0.988i$
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  + (0.951 − 0.309i)2-s + (−0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (0.618 + 2.14i)5-s − 0.999·6-s + (1.68 + 2.31i)7-s + (0.587 − 0.809i)8-s + (0.809 + 0.587i)9-s + (1.25 + 1.85i)10-s + (−4.86 + 3.53i)11-s + (−0.951 + 0.309i)12-s + (0.678 + 0.220i)13-s + (2.31 + 1.68i)14-s + (0.0762 − 2.23i)15-s + (0.309 − 0.951i)16-s + (−1.19 + 1.64i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (−0.549 − 0.178i)3-s + (0.404 − 0.293i)4-s + (0.276 + 0.961i)5-s − 0.408·6-s + (0.635 + 0.875i)7-s + (0.207 − 0.286i)8-s + (0.269 + 0.195i)9-s + (0.395 + 0.585i)10-s + (−1.46 + 1.06i)11-s + (−0.274 + 0.0892i)12-s + (0.188 + 0.0611i)13-s + (0.618 + 0.449i)14-s + (0.0196 − 0.577i)15-s + (0.0772 − 0.237i)16-s + (−0.289 + 0.398i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.149 - 0.988i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.149 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32389 + 1.13843i\)
\(L(\frac12)\) \(\approx\) \(1.32389 + 1.13843i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-0.618 - 2.14i)T \)
31 \( 1 + (5.55 + 0.436i)T \)
good7 \( 1 + (-1.68 - 2.31i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (4.86 - 3.53i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.678 - 0.220i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.19 - 1.64i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.57 + 4.84i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.29 - 4.53i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.71 - 5.26i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 + (0.325 + 1.00i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-8.96 + 2.91i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-11.0 - 3.58i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.79 + 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.82 + 5.61i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 11.3iT - 67T^{2} \)
71 \( 1 + (-1.32 - 0.963i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.11 - 8.41i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.63 + 4.09i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.8 - 3.51i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.4 - 8.32i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.63 + 3.62i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.50779806592483076132181423826, −9.741384753877573578456161691759, −8.511155194714637188321611574895, −7.38520975634173342499008958932, −6.84617326526477594319040507770, −5.59753722960107329327342817589, −5.28683045489341872906535562927, −4.07132074831383472757772620383, −2.59403128001561216121474986358, −2.00666881571382954501861567107, 0.67639902764296873455600413727, 2.31768659132229590534476838644, 3.93905520635192954865717788966, 4.56360340637241343432706205165, 5.61797793036156106247124658021, 5.94978495416612380173065506463, 7.41766073434458753968282858001, 8.054300205730303458467283316274, 8.873404578687611964437411235214, 10.25082706052408927057202193274

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