↑

Properties

Label 2-930-31.8-c1-0-9
Degree $2$
Conductor $930$
Sign $0.880 - 0.473i$
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

Related objects

Downloads

Learn more

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 + (3.29 + 2.39i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.686 − 0.499i)11-s + (0.309 − 0.951i)12-s + (1.59 + 4.91i)13-s + (3.29 − 2.39i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−2.38 + 1.73i)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 + (1.24 + 0.904i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (0.0977 − 0.300i)10-s + (−0.207 − 0.150i)11-s + (0.0892 − 0.274i)12-s + (0.442 + 1.36i)13-s + (0.880 − 0.639i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.578 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.880 - 0.473i$
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: \((2,\ 930,\ (\ :1/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01613 + 0.507227i\)
\(L(\frac12)\) \(\approx\) \(2.01613 + 0.507227i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.51 + 0.782i)T \)
good7 \( 1 + (-3.29 - 2.39i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.686 + 0.499i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.59 - 4.91i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.38 - 1.73i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.145 + 0.449i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (6.82 - 4.96i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-3.00 + 9.25i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 3.34T + 37T^{2} \)
41 \( 1 + (2.43 - 7.48i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.999 + 3.07i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-2.90 - 8.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.34 + 6.79i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.53 + 4.73i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 5.10T + 67T^{2} \)
71 \( 1 + (-11.2 + 8.19i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.70 + 4.87i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.92 + 1.40i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.41 - 10.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (12.6 + 9.17i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-12.4 - 9.07i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06285359056924159355746015876, −9.455394099976403487141151380940, −8.554194540368187391766084251872, −8.048340721636177658381801783155, −6.40766337901117373071977282946, −5.60659824851979634526391847292, −4.65727887886748442069643722941, −3.95028530623281193900605555273, −2.44342305303580175130099628152, −1.74611828965903320645755633797, 0.960732709468419276355907614760, 2.41317832682936900996031736172, 3.82474635611809909846689907199, 4.89215527836227953961243826080, 5.64286317110906934479576085930, 6.74301471147477901362145766120, 7.40731019126537262566113807245, 8.320735879591875803389525027062, 8.651364413727478030363100839499, 10.23902884616779495033062480038

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