↑

Properties

Label 2-507-39.8-c1-0-19
Degree $2$
Conductor $507$
Sign $0.576 - 0.816i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−1.06 + 1.06i)2-s + (1.36 − 1.06i)3-s − 0.267i·4-s + (−1.06 + 1.06i)5-s + (−0.320 + 2.58i)6-s + (1 − i)7-s + (−1.84 − 1.84i)8-s + (0.732 − 2.90i)9-s − 2.26i·10-s + (2.90 + 2.90i)11-s + (−0.285 − 0.366i)12-s + 2.12i·14-s + (−0.320 + 2.58i)15-s + 4.46·16-s + 5.03·17-s + (2.31 + 3.87i)18-s + ⋯
L(s)  = 1  + (−0.752 + 0.752i)2-s + (0.788 − 0.614i)3-s − 0.133i·4-s + (−0.476 + 0.476i)5-s + (−0.130 + 1.05i)6-s + (0.377 − 0.377i)7-s + (−0.652 − 0.652i)8-s + (0.244 − 0.969i)9-s − 0.717i·10-s + (0.877 + 0.877i)11-s + (−0.0823 − 0.105i)12-s + 0.569i·14-s + (−0.0827 + 0.668i)15-s + 1.11·16-s + 1.22·17-s + (0.546 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.576 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.576 - 0.816i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.576 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11578 + 0.578161i\)
\(L(\frac12)\) \(\approx\) \(1.11578 + 0.578161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 + 1.06i)T \)
13 \( 1 \)
good2 \( 1 + (1.06 - 1.06i)T - 2iT^{2} \)
5 \( 1 + (1.06 - 1.06i)T - 5iT^{2} \)
7 \( 1 + (-1 + i)T - 7iT^{2} \)
11 \( 1 + (-2.90 - 2.90i)T + 11iT^{2} \)
17 \( 1 - 5.03T + 17T^{2} \)
19 \( 1 + (-2.73 - 2.73i)T + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 7.16iT - 29T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (-3.83 + 3.83i)T - 37iT^{2} \)
41 \( 1 + (-3.97 + 3.97i)T - 41iT^{2} \)
43 \( 1 + 2.19iT - 43T^{2} \)
47 \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \)
53 \( 1 - 0.779iT - 53T^{2} \)
59 \( 1 + (2.12 + 2.12i)T + 59iT^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + (4.19 + 4.19i)T + 67iT^{2} \)
71 \( 1 + (-2.12 + 2.12i)T - 71iT^{2} \)
73 \( 1 + (0.901 - 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (2.90 - 2.90i)T - 83iT^{2} \)
89 \( 1 + (-6.59 - 6.59i)T + 89iT^{2} \)
97 \( 1 + (1.19 + 1.19i)T + 97iT^{2} \)
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

−10.94151027608129954596271565802, −9.675589457602507541054471756937, −9.184276119209354296298416316345, −8.075046730397594945233359004657, −7.40513252444704436697084794019, −7.09725546552932108661109833772, −5.88085328655945960545009465693, −4.02389652927999046409228721409, −3.18162726785623306526391600049, −1.35076656734382330975805788655, 1.11615308883659204162559493260, 2.61635569794629682528658338350, 3.67779179707514730927934544636, 4.91222196436023666646654342828, 5.98223016964476905085784805715, 7.73759057784392467309420030931, 8.412623485003661459375140724116, 9.100339159573535743334831773723, 9.755279737971344273211734801785, 10.64082978369083238938258292038

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