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Properties

Label 2-930-31.4-c1-0-10
Degree $2$
Conductor $930$
Sign $0.155 - 0.987i$
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 + (2.14 − 1.55i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (3.05 − 2.21i)11-s + (−0.309 − 0.951i)12-s + (0.0808 − 0.248i)13-s + (2.14 + 1.55i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (2.72 + 1.97i)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 + (0.810 − 0.588i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (0.920 − 0.668i)11-s + (−0.0892 − 0.274i)12-s + (0.0224 − 0.0690i)13-s + (0.572 + 0.416i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.660 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29075 + 1.10292i\)
\(L(\frac12)\) \(\approx\) \(1.29075 + 1.10292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 5.46i)T \)
good7 \( 1 + (-2.14 + 1.55i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-3.05 + 2.21i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.0808 + 0.248i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.72 - 1.97i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.47 - 4.54i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.19 - 4.50i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.38 - 4.26i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 6.85T + 37T^{2} \)
41 \( 1 + (0.633 + 1.95i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.18 + 6.71i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.326 - 1.00i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.26 - 6.00i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.46 - 4.51i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 1.91T + 61T^{2} \)
67 \( 1 - 4.02T + 67T^{2} \)
71 \( 1 + (0.666 + 0.484i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.91 + 5.02i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.06 - 2.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.49 - 10.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (4.84 - 3.52i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.68 + 4.13i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.31091491158299508404467701395, −9.245672192095728466390167364257, −8.505480989964551549631321032732, −7.69331460182352819514454522893, −6.92142378019166758550111983406, −5.78031723862466319151079514057, −5.09040223018446219943157643137, −3.95530321982776820816317323735, −3.45712992102763033544824842853, −1.20271634015934229036967616440, 1.00682725906023500824306679595, 2.20710792977011088020890157688, 3.33171917546674112309611113451, 4.68897407743764599494266672208, 5.17651804977963055435846141061, 6.55284531234219221114181379803, 7.25681999862421694188510674846, 8.378790510649353269475882478185, 8.988444821792969014771293106387, 9.945228801884233157169125250511

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