↑

Properties

Label 2-930-31.4-c1-0-4
Degree $2$
Conductor $930$
Sign $0.886 - 0.462i$
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 + (−1.96 + 1.42i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−1.46 + 1.06i)11-s + (0.309 + 0.951i)12-s + (−1.45 + 4.48i)13-s + (1.96 + 1.42i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (4.05 + 2.94i)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 + (−0.740 + 0.538i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (−0.440 + 0.319i)11-s + (0.0892 + 0.274i)12-s + (−0.404 + 1.24i)13-s + (0.523 + 0.380i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.983 + 0.714i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.886 - 0.462i$
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: \((2,\ 930,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04462 + 0.255958i\)
\(L(\frac12)\) \(\approx\) \(1.04462 + 0.255958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.74 - 4.11i)T \)
good7 \( 1 + (1.96 - 1.42i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.46 - 1.06i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.45 - 4.48i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.05 - 2.94i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.100 + 0.308i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.902 + 0.655i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.50 - 7.71i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 8.55T + 37T^{2} \)
41 \( 1 + (0.649 + 1.99i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (1.66 + 5.13i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.86 - 5.72i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.13 + 0.825i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.07 - 6.38i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 - 16.3T + 67T^{2} \)
71 \( 1 + (-8.91 - 6.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.87 + 3.53i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.44 + 2.50i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.69 + 8.30i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.37 - 5.35i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-14.2 + 10.3i)T + (29.9 - 92.2i)T^{2} \)
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

−10.04195230538863545811749751834, −9.335839937426961738499408278661, −8.709340715653112437483198318014, −7.66236619808058270339054207124, −6.76217245813012912305426586524, −5.87802583174373457173203526699, −4.79341986272910992868146072989, −3.46842973483818358091276272135, −2.50559028008773312276225003735, −1.50614150671055380684600189251, 0.54245948703093390254786453757, 2.69049076370898106990828189550, 3.66807936986109564745746326243, 4.92019325775309357733983647335, 5.69557555771584012437531418128, 6.49317708928168030238585959854, 7.73860922185132665271961013420, 8.056071893671025485296860948081, 9.473818321968955339110671084774, 9.788199315578946449327433127571

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