↑

Properties

Label 2-2385-265.124-c0-0-0
Degree $2$
Conductor $2385$
Sign $0.104 - 0.994i$
Analytic cond. $1.19027$
Root an. cond. $1.09099$
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  + (−1.40 − 0.851i)2-s + (0.794 + 1.51i)4-s + (−0.911 − 0.410i)5-s + (0.0704 − 1.16i)8-s + (0.935 + 1.35i)10-s + (−0.120 + 0.174i)16-s + (0.269 + 0.239i)17-s + (−1.17 + 0.366i)19-s + (−0.103 − 1.70i)20-s + (−0.170 + 0.170i)23-s + (0.663 + 0.748i)25-s + (−1.34 − 1.05i)31-s + (−0.746 + 0.335i)32-s + (−0.176 − 0.566i)34-s + (1.97 + 0.485i)38-s + ⋯
L(s)  = 1  + (−1.40 − 0.851i)2-s + (0.794 + 1.51i)4-s + (−0.911 − 0.410i)5-s + (0.0704 − 1.16i)8-s + (0.935 + 1.35i)10-s + (−0.120 + 0.174i)16-s + (0.269 + 0.239i)17-s + (−1.17 + 0.366i)19-s + (−0.103 − 1.70i)20-s + (−0.170 + 0.170i)23-s + (0.663 + 0.748i)25-s + (−1.34 − 1.05i)31-s + (−0.746 + 0.335i)32-s + (−0.176 − 0.566i)34-s + (1.97 + 0.485i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $0.104 - 0.994i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ 0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1267848454\)
\(L(\frac12)\) \(\approx\) \(0.1267848454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.911 + 0.410i)T \)
53 \( 1 + (-0.0603 - 0.998i)T \)
good2 \( 1 + (1.40 + 0.851i)T + (0.464 + 0.885i)T^{2} \)
7 \( 1 + (0.885 - 0.464i)T^{2} \)
11 \( 1 + (0.970 + 0.239i)T^{2} \)
13 \( 1 + (-0.568 - 0.822i)T^{2} \)
17 \( 1 + (-0.269 - 0.239i)T + (0.120 + 0.992i)T^{2} \)
19 \( 1 + (1.17 - 0.366i)T + (0.822 - 0.568i)T^{2} \)
23 \( 1 + (0.170 - 0.170i)T - iT^{2} \)
29 \( 1 + (0.970 - 0.239i)T^{2} \)
31 \( 1 + (1.34 + 1.05i)T + (0.239 + 0.970i)T^{2} \)
37 \( 1 + (-0.354 - 0.935i)T^{2} \)
41 \( 1 + (-0.239 + 0.970i)T^{2} \)
43 \( 1 + (-0.354 + 0.935i)T^{2} \)
47 \( 1 + (0.556 + 0.210i)T + (0.748 + 0.663i)T^{2} \)
59 \( 1 + (0.748 + 0.663i)T^{2} \)
61 \( 1 + (0.120 + 0.00729i)T + (0.992 + 0.120i)T^{2} \)
67 \( 1 + (-0.822 - 0.568i)T^{2} \)
71 \( 1 + (-0.935 - 0.354i)T^{2} \)
73 \( 1 + (-0.992 + 0.120i)T^{2} \)
79 \( 1 + (1.56 - 0.943i)T + (0.464 - 0.885i)T^{2} \)
83 \( 1 + (-0.657 - 0.657i)T + iT^{2} \)
89 \( 1 + (0.120 + 0.992i)T^{2} \)
97 \( 1 + (0.748 - 0.663i)T^{2} \)
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

−9.230621518419971869305645252615, −8.722068405862252629030571538792, −7.922522555209376297619584411332, −7.58871280368738840867389836816, −6.51164636165695945018958817082, −5.37370138634795474347795209211, −4.20601554332691400308035640035, −3.48002552645656970309135117692, −2.34782142786850660196610659072, −1.30669633793806081000057514625, 0.14504643246485794346885214783, 1.74106156784258619109237606992, 3.12926103676406730124282761291, 4.17077287105807154577116802374, 5.25556879712805997574564764467, 6.32667615211836541978799209656, 6.90029979956585205179757411821, 7.50665601848430782879308879560, 8.275528283108012105617309778684, 8.721049391103953318165268937155

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