↑

Properties

Label 2-930-155.4-c1-0-8
Degree $2$
Conductor $930$
Sign $0.867 - 0.497i$
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  + (−0.951 + 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 − 0.587i)4-s + (−1.82 − 1.28i)5-s − 0.999·6-s + (0.907 + 1.24i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (2.13 + 0.657i)10-s + (−0.748 + 0.543i)11-s + (0.951 − 0.309i)12-s + (0.363 + 0.118i)13-s + (−1.24 − 0.907i)14-s + (−1.34 − 1.78i)15-s + (0.309 − 0.951i)16-s + (2.98 − 4.11i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.549 + 0.178i)3-s + (0.404 − 0.293i)4-s + (−0.818 − 0.575i)5-s − 0.408·6-s + (0.342 + 0.471i)7-s + (−0.207 + 0.286i)8-s + (0.269 + 0.195i)9-s + (0.675 + 0.207i)10-s + (−0.225 + 0.163i)11-s + (0.274 − 0.0892i)12-s + (0.100 + 0.0327i)13-s + (−0.333 − 0.242i)14-s + (−0.346 − 0.461i)15-s + (0.0772 − 0.237i)16-s + (0.725 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.867 - 0.497i$
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: \((2,\ 930,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22552 + 0.326622i\)
\(L(\frac12)\) \(\approx\) \(1.22552 + 0.326622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.82 + 1.28i)T \)
31 \( 1 + (0.474 - 5.54i)T \)
good7 \( 1 + (-0.907 - 1.24i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.748 - 0.543i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.363 - 0.118i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.98 + 4.11i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.91 - 5.88i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.62 + 3.61i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.226 - 0.697i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 2.46iT - 37T^{2} \)
41 \( 1 + (-1.17 - 3.62i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-9.18 + 2.98i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-9.29 - 3.01i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.20 + 5.78i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.32 - 10.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 8.96iT - 67T^{2} \)
71 \( 1 + (2.77 + 2.01i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.32 + 8.70i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (5.01 + 3.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.10 - 1.98i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.19 + 4.50i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-10.0 - 13.7i)T + (-29.9 + 92.2i)T^{2} \)
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

−10.00900996893567260075456787488, −9.015471225628397879012926015921, −8.599858966676853057938666538914, −7.68530340509792212408546269355, −7.21235885652342241285269670389, −5.72991498421740076227014047036, −4.91543037162556088038162596751, −3.76004686259046850071966700028, −2.59019894442369876094317361326, −1.09989212043934704305406359614, 0.915615303748078943144159539009, 2.48251511766173790427676232843, 3.44692861539880352794254545053, 4.33968285407137559345769929321, 5.83833698438741569543006464226, 7.09110920601746392204481106026, 7.52654843937444560018874019912, 8.258195884026691190438713259108, 9.087802787715839118322610336145, 10.00589546698213962934060316311

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