↑

Properties

Label 2-1265-1265.263-c0-0-1
Degree $2$
Conductor $1265$
Sign $-0.100 + 0.994i$
Analytic cond. $0.631317$
Root an. cond. $0.794554$
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  + (−0.203 − 0.936i)3-s + (0.989 + 0.142i)4-s + (−0.281 − 0.959i)5-s + (0.0739 − 0.0337i)9-s + (−0.755 − 0.654i)11-s + (−0.0683 − 0.956i)12-s + (−0.841 + 0.459i)15-s + (0.959 + 0.281i)16-s + (−0.142 − 0.989i)20-s + (−0.654 + 0.755i)23-s + (−0.841 + 0.540i)25-s + (−0.621 − 0.829i)27-s + (1.53 − 0.983i)31-s + (−0.459 + 0.841i)33-s + (0.0779 − 0.0228i)36-s + (−0.559 + 1.50i)37-s + ⋯
L(s)  = 1  + (−0.203 − 0.936i)3-s + (0.989 + 0.142i)4-s + (−0.281 − 0.959i)5-s + (0.0739 − 0.0337i)9-s + (−0.755 − 0.654i)11-s + (−0.0683 − 0.956i)12-s + (−0.841 + 0.459i)15-s + (0.959 + 0.281i)16-s + (−0.142 − 0.989i)20-s + (−0.654 + 0.755i)23-s + (−0.841 + 0.540i)25-s + (−0.621 − 0.829i)27-s + (1.53 − 0.983i)31-s + (−0.459 + 0.841i)33-s + (0.0779 − 0.0228i)36-s + (−0.559 + 1.50i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164853910\)
\(L(\frac12)\) \(\approx\) \(1.164853910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.281 + 0.959i)T \)
11 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
good2 \( 1 + (-0.989 - 0.142i)T^{2} \)
3 \( 1 + (0.203 + 0.936i)T + (-0.909 + 0.415i)T^{2} \)
7 \( 1 + (0.540 - 0.841i)T^{2} \)
13 \( 1 + (-0.540 - 0.841i)T^{2} \)
17 \( 1 + (0.281 + 0.959i)T^{2} \)
19 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-1.53 + 0.983i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 + (0.559 - 1.50i)T + (-0.755 - 0.654i)T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (-0.909 + 0.415i)T^{2} \)
47 \( 1 + (-1.32 + 1.32i)T - iT^{2} \)
53 \( 1 + (1.05 - 0.574i)T + (0.540 - 0.841i)T^{2} \)
59 \( 1 + (-0.557 - 1.89i)T + (-0.841 + 0.540i)T^{2} \)
61 \( 1 + (0.415 - 0.909i)T^{2} \)
67 \( 1 + (-0.125 + 1.75i)T + (-0.989 - 0.142i)T^{2} \)
71 \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \)
73 \( 1 + (-0.281 + 0.959i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.755 - 0.654i)T^{2} \)
89 \( 1 + (-0.474 - 0.304i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.654 + 0.244i)T + (0.755 - 0.654i)T^{2} \)
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

−9.731484069665163738201560081327, −8.542577807450033725652765909873, −7.86307171435758997214595569375, −7.38456202536336910396309329437, −6.29242239339032615561994998844, −5.77213771227629941095926539189, −4.58814720584317055584000289206, −3.35536722081293910801061144544, −2.15073227690020646821258623075, −1.07011833281080179796373278089, 2.07295873061305068212832199125, 2.98851161500387249090877010557, 4.02559663335109974138562976514, 4.99126869825237686406725981944, 5.98347049745477938226913819374, 6.86744973273129229033015568396, 7.48352737282798278394091047918, 8.354452274413599345978986019696, 9.753345821534384314085976046548, 10.22328258587090158637384485243

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