↑

Properties

Label 2-507-13.12-c3-0-50
Degree $2$
Conductor $507$
Sign $0.832 - 0.554i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s + 3·3-s − 4-s − 9i·5-s + 9i·6-s − 2i·7-s + 21i·8-s + 9·9-s + 27·10-s − 30i·11-s − 3·12-s + 6·14-s − 27i·15-s − 71·16-s + 111·17-s + 27i·18-s + ⋯
L(s)  = 1  + 1.06i·2-s + 0.577·3-s − 0.125·4-s − 0.804i·5-s + 0.612i·6-s − 0.107i·7-s + 0.928i·8-s + 0.333·9-s + 0.853·10-s − 0.822i·11-s − 0.0721·12-s + 0.114·14-s − 0.464i·15-s − 1.10·16-s + 1.58·17-s + 0.353i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 - 0.554i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.827701846\)
\(L(\frac12)\) \(\approx\) \(2.827701846\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 - 3iT - 8T^{2} \)
5 \( 1 + 9iT - 125T^{2} \)
7 \( 1 + 2iT - 343T^{2} \)
11 \( 1 + 30iT - 1.33e3T^{2} \)
17 \( 1 - 111T + 4.91e3T^{2} \)
19 \( 1 + 46iT - 6.85e3T^{2} \)
23 \( 1 - 6T + 1.21e4T^{2} \)
29 \( 1 + 105T + 2.43e4T^{2} \)
31 \( 1 + 100iT - 2.97e4T^{2} \)
37 \( 1 + 17iT - 5.06e4T^{2} \)
41 \( 1 + 231iT - 6.89e4T^{2} \)
43 \( 1 - 514T + 7.95e4T^{2} \)
47 \( 1 - 162iT - 1.03e5T^{2} \)
53 \( 1 - 639T + 1.48e5T^{2} \)
59 \( 1 + 600iT - 2.05e5T^{2} \)
61 \( 1 - 233T + 2.26e5T^{2} \)
67 \( 1 - 926iT - 3.00e5T^{2} \)
71 \( 1 + 930iT - 3.57e5T^{2} \)
73 \( 1 - 253iT - 3.89e5T^{2} \)
79 \( 1 + 1.32e3T + 4.93e5T^{2} \)
83 \( 1 - 810iT - 5.71e5T^{2} \)
89 \( 1 + 498iT - 7.04e5T^{2} \)
97 \( 1 - 1.35e3iT - 9.12e5T^{2} \)
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

−10.49278502485980065516205231771, −9.271044832754378402893121260786, −8.636838522435431763154809929807, −7.82018965672483181596001548996, −7.13617338298115841646644137761, −5.87141184929533907082343479485, −5.24411001507637250431035731266, −3.92352572845401671204318336114, −2.56718115602339167860997700036, −0.942553052629613768336905339397, 1.25107733736386298459950918832, 2.42109905357614104623334134488, 3.23764905901582361463887967081, 4.17304356705798184224953442369, 5.73470371751573264762651872308, 7.02184628756134408776418005813, 7.57204882251368678362122128521, 8.896321945762358806975591332684, 9.952875142639320099087530157564, 10.27678623868498215438542083979

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