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Properties

Label 2-930-31.8-c1-0-18
Degree $2$
Conductor $930$
Sign $-0.225 + 0.974i$
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 + (3.89 + 2.82i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−5.16 − 3.75i)11-s + (−0.309 + 0.951i)12-s + (−0.206 − 0.636i)13-s + (3.89 − 2.82i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (4.68 − 3.40i)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 + (1.47 + 1.06i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (0.0977 − 0.300i)10-s + (−1.55 − 1.13i)11-s + (−0.0892 + 0.274i)12-s + (−0.0573 − 0.176i)13-s + (1.04 − 0.755i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (1.13 − 0.824i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.225 + 0.974i$
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: \((2,\ 930,\ (\ :1/2),\ -0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15437 - 1.45171i\)
\(L(\frac12)\) \(\approx\) \(1.15437 - 1.45171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 + 4.65i)T \)
good7 \( 1 + (-3.89 - 2.82i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (5.16 + 3.75i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.206 + 0.636i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.68 + 3.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.409 - 1.25i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.26 + 4.55i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.718 + 2.21i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 1.94T + 37T^{2} \)
41 \( 1 + (-1.23 + 3.79i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-1.79 + 5.51i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.567 - 1.74i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.03 - 5.11i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.09 + 6.45i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 1.85T + 61T^{2} \)
67 \( 1 + 6.44T + 67T^{2} \)
71 \( 1 + (8.01 - 5.82i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.92 - 2.84i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (6.90 - 5.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (4.55 - 14.0i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.21 - 4.51i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.9 + 7.96i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.02198978515672688096754576647, −8.872032851276138927431958217974, −8.232420216090207381095906002595, −7.54762662623305284370798630912, −5.94801202413986359932888102345, −5.42763981095645264933346712307, −4.77313311338779479933864824585, −2.91018645965406846883360807628, −2.34793751716125857980694564484, −0.937153360723023866621436406321, 1.48518380780191616981374348183, 3.14433418389959805338643974674, 4.59684533256976589280555926721, 4.86617774193580385857300359736, 5.76118630369455600313208283657, 7.12475758267279630166215008307, 7.66797402374737272676456703549, 8.403600151516859279927266797787, 9.569291434092444019905265201797, 10.38322019531757735694991241873

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