↑

Properties

Label 2-507-13.12-c3-0-45
Degree $2$
Conductor $507$
Sign $0.999 + 0.0304i$
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  + 1.61i·2-s − 3·3-s + 5.39·4-s + 1.20i·5-s − 4.84i·6-s − 28.2i·7-s + 21.6i·8-s + 9·9-s − 1.95·10-s + 31.7i·11-s − 16.1·12-s + 45.6·14-s − 3.62i·15-s + 8.22·16-s + 16.0·17-s + 14.5i·18-s + ⋯
L(s)  = 1  + 0.570i·2-s − 0.577·3-s + 0.674·4-s + 0.108i·5-s − 0.329i·6-s − 1.52i·7-s + 0.955i·8-s + 0.333·9-s − 0.0617·10-s + 0.871i·11-s − 0.389·12-s + 0.871·14-s − 0.0624i·15-s + 0.128·16-s + 0.229·17-s + 0.190i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.999 + 0.0304i$
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: \((2,\ 507,\ (\ :3/2),\ 0.999 + 0.0304i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.934231881\)
\(L(\frac12)\) \(\approx\) \(1.934231881\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.61iT - 8T^{2} \)
5 \( 1 - 1.20iT - 125T^{2} \)
7 \( 1 + 28.2iT - 343T^{2} \)
11 \( 1 - 31.7iT - 1.33e3T^{2} \)
17 \( 1 - 16.0T + 4.91e3T^{2} \)
19 \( 1 + 58.5iT - 6.85e3T^{2} \)
23 \( 1 + 152.T + 1.21e4T^{2} \)
29 \( 1 - 265.T + 2.43e4T^{2} \)
31 \( 1 + 56.9iT - 2.97e4T^{2} \)
37 \( 1 + 444. iT - 5.06e4T^{2} \)
41 \( 1 + 189. iT - 6.89e4T^{2} \)
43 \( 1 - 132.T + 7.95e4T^{2} \)
47 \( 1 + 113. iT - 1.03e5T^{2} \)
53 \( 1 - 300.T + 1.48e5T^{2} \)
59 \( 1 + 513. iT - 2.05e5T^{2} \)
61 \( 1 - 619.T + 2.26e5T^{2} \)
67 \( 1 - 597. iT - 3.00e5T^{2} \)
71 \( 1 - 826. iT - 3.57e5T^{2} \)
73 \( 1 - 332. iT - 3.89e5T^{2} \)
79 \( 1 - 679.T + 4.93e5T^{2} \)
83 \( 1 - 88.2iT - 5.71e5T^{2} \)
89 \( 1 + 1.48e3iT - 7.04e5T^{2} \)
97 \( 1 - 154. iT - 9.12e5T^{2} \)
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

−10.49302404128564823202626566964, −9.950430933779188232428861468395, −8.438938881020224550330057674379, −7.27380995112965577334981194298, −7.07997457666007917826297670751, −6.05168046924597855769100062477, −4.89682378385575994876874260552, −3.90920077694078948443566226304, −2.26217219658794537656915497252, −0.75269549250615826138947308949, 1.09793783484110532465089841422, 2.39492205465640170020628027950, 3.36292907043067454166655762264, 4.91021921307356989545467052428, 6.04147260546235568576640073411, 6.43088718348696531738971612658, 7.939181507871091915085114005942, 8.747985452303774185347691939876, 9.910921857960237137227988696082, 10.57295558299730987296765186419

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