↑

Properties

Label 2-3380-3380.2779-c0-0-0
Degree $2$
Conductor $3380$
Sign $0.439 + 0.898i$
Analytic cond. $1.68683$
Root an. cond. $1.29878$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.278 + 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.987 + 0.160i)5-s + (−0.748 − 0.663i)8-s + (−0.692 + 0.721i)9-s + (−0.428 − 0.903i)10-s + (−0.948 − 0.316i)13-s + (0.428 − 0.903i)16-s + (0.338 + 1.01i)17-s + (−0.885 − 0.464i)18-s + (0.748 − 0.663i)20-s + (0.948 − 0.316i)25-s + (0.0402 − 0.999i)26-s + (−0.444 − 1.53i)29-s + (0.987 + 0.160i)32-s + ⋯
L(s)  = 1  + (0.278 + 0.960i)2-s + (−0.845 + 0.534i)4-s + (−0.987 + 0.160i)5-s + (−0.748 − 0.663i)8-s + (−0.692 + 0.721i)9-s + (−0.428 − 0.903i)10-s + (−0.948 − 0.316i)13-s + (0.428 − 0.903i)16-s + (0.338 + 1.01i)17-s + (−0.885 − 0.464i)18-s + (0.748 − 0.663i)20-s + (0.948 − 0.316i)25-s + (0.0402 − 0.999i)26-s + (−0.444 − 1.53i)29-s + (0.987 + 0.160i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3380\)    =    \(2^{2} \cdot 5 \cdot 13^{2}\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(1.68683\)
Root analytic conductor: \(1.29878\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3380} (2779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3380,\ (\ :0),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1667141767\)
\(L(\frac12)\) \(\approx\) \(0.1667141767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.278 - 0.960i)T \)
5 \( 1 + (0.987 - 0.160i)T \)
13 \( 1 + (0.948 + 0.316i)T \)
good3 \( 1 + (0.692 - 0.721i)T^{2} \)
7 \( 1 + (-0.799 + 0.600i)T^{2} \)
11 \( 1 + (-0.0402 + 0.999i)T^{2} \)
17 \( 1 + (-0.338 - 1.01i)T + (-0.799 + 0.600i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.444 + 1.53i)T + (-0.845 + 0.534i)T^{2} \)
31 \( 1 + (-0.748 - 0.663i)T^{2} \)
37 \( 1 + (1.96 - 0.319i)T + (0.948 - 0.316i)T^{2} \)
41 \( 1 + (0.752 + 1.76i)T + (-0.692 + 0.721i)T^{2} \)
43 \( 1 + (0.948 + 0.316i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
53 \( 1 + (-0.419 + 0.474i)T + (-0.120 - 0.992i)T^{2} \)
59 \( 1 + (-0.632 + 0.774i)T^{2} \)
61 \( 1 + (0.368 + 1.80i)T + (-0.919 + 0.391i)T^{2} \)
67 \( 1 + (-0.428 - 0.903i)T^{2} \)
71 \( 1 + (0.278 + 0.960i)T^{2} \)
73 \( 1 + (-0.970 + 0.239i)T + (0.885 - 0.464i)T^{2} \)
79 \( 1 + (-0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.970 - 0.239i)T^{2} \)
89 \( 1 + (1.42 - 0.822i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.706 - 0.0570i)T + (0.987 - 0.160i)T^{2} \)
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

−8.291387595881148666031272943305, −7.961748105040940608099479928920, −7.22930405133155317025847843709, −6.54227228652508366438719091541, −5.48570782381295350691436054578, −5.07496689188134376758763852507, −4.02128021744286600913196237710, −3.42996633478285965219785920380, −2.28944758650241998780552289367, −0.094361969008589597920067024229, 1.30071235969159395296220918102, 2.74822832753227274427933557894, 3.28494744327164450304126402644, 4.16434346667366847720300826606, 4.98705994161441725389636958031, 5.56768445472628474180919887650, 6.79506420951404805102267851933, 7.42078614258606533263974385578, 8.495313464418408368556841720451, 8.949336096907608772479095529683

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