Properties

Label 2-1500-25.4-c1-0-5
Degree $2$
Conductor $1500$
Sign $0.626 - 0.779i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (−0.587 + 0.809i)3-s + 2.44i·7-s + (−0.309 − 0.951i)9-s + (−0.178 + 0.548i)11-s + (6.13 − 1.99i)13-s + (−1.11 − 1.53i)17-s + (6.69 − 4.86i)19-s + (−1.97 − 1.43i)21-s + (−4.00 − 1.30i)23-s + (0.951 + 0.309i)27-s + (5.28 + 3.84i)29-s + (−3.93 + 2.86i)31-s + (−0.339 − 0.466i)33-s + (0.207 − 0.0673i)37-s + (−1.99 + 6.13i)39-s + ⋯
L(s)  = 1  + (−0.339 + 0.467i)3-s + 0.923i·7-s + (−0.103 − 0.317i)9-s + (−0.0537 + 0.165i)11-s + (1.70 − 0.552i)13-s + (−0.270 − 0.372i)17-s + (1.53 − 1.11i)19-s + (−0.431 − 0.313i)21-s + (−0.834 − 0.271i)23-s + (0.183 + 0.0594i)27-s + (0.982 + 0.713i)29-s + (−0.707 + 0.513i)31-s + (−0.0590 − 0.0812i)33-s + (0.0340 − 0.0110i)37-s + (−0.318 + 0.981i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.604950760\)
\(L(\frac12)\) \(\approx\) \(1.604950760\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + (0.178 - 0.548i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-6.13 + 1.99i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.11 + 1.53i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-6.69 + 4.86i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.00 + 1.30i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-5.28 - 3.84i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.93 - 2.86i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.207 + 0.0673i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.99 - 6.13i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.42iT - 43T^{2} \)
47 \( 1 + (5.65 - 7.78i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-8.22 + 11.3i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.72 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.48 - 4.57i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (2.27 + 3.13i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.24 - 3.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.55 - 3.10i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.83 + 2.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.01 - 2.77i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.75 + 5.40i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.87 + 5.32i)T + (-29.9 - 92.2i)T^{2} \)
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

−9.540370799653936579918196338155, −8.839315168977483688579962861218, −8.219813094257407310303730723041, −7.07821029671349600281873114479, −6.17184570017644215986939975505, −5.47851658648084908310864509857, −4.71176650216493062700742283285, −3.52499104409256172115993439055, −2.69371618933210722815033739899, −1.09363230242419340920750577600, 0.864581542224286834459226945186, 1.89065029620668100807482794990, 3.55935127932640596225776852998, 4.06506371908850278619830247146, 5.44170109000547116847644022937, 6.12557558294005937063032464416, 6.91150189116186109005774346988, 7.78281327606699465656214745912, 8.377451363558147133535370313741, 9.405087587494899318593954350818

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