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Properties

Label 2-1265-1265.43-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.950 - 0.311i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.83 − 0.682i)3-s + (0.281 + 0.959i)4-s + (−0.540 + 0.841i)5-s + (2.13 − 1.84i)9-s + (−0.989 − 0.142i)11-s + (1.17 + 1.56i)12-s + (−0.415 + 1.90i)15-s + (−0.841 + 0.540i)16-s + (−0.959 − 0.281i)20-s + (−0.142 + 0.989i)23-s + (−0.415 − 0.909i)25-s + (1.70 − 3.12i)27-s + (−0.627 − 1.37i)31-s + (−1.90 + 0.415i)33-s + (2.37 + 1.52i)36-s + (0.125 − 1.75i)37-s + ⋯
L(s)  = 1  + (1.83 − 0.682i)3-s + (0.281 + 0.959i)4-s + (−0.540 + 0.841i)5-s + (2.13 − 1.84i)9-s + (−0.989 − 0.142i)11-s + (1.17 + 1.56i)12-s + (−0.415 + 1.90i)15-s + (−0.841 + 0.540i)16-s + (−0.959 − 0.281i)20-s + (−0.142 + 0.989i)23-s + (−0.415 − 0.909i)25-s + (1.70 − 3.12i)27-s + (−0.627 − 1.37i)31-s + (−1.90 + 0.415i)33-s + (2.37 + 1.52i)36-s + (0.125 − 1.75i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.950 - 0.311i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.950 - 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.802260818\)
\(L(\frac12)\) \(\approx\) \(1.802260818\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.540 - 0.841i)T \)
11 \( 1 + (0.989 + 0.142i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
good2 \( 1 + (-0.281 - 0.959i)T^{2} \)
3 \( 1 + (-1.83 + 0.682i)T + (0.755 - 0.654i)T^{2} \)
7 \( 1 + (-0.909 - 0.415i)T^{2} \)
13 \( 1 + (0.909 - 0.415i)T^{2} \)
17 \( 1 + (0.540 - 0.841i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.627 + 1.37i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (-0.125 + 1.75i)T + (-0.989 - 0.142i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.755 - 0.654i)T^{2} \)
47 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
53 \( 1 + (0.203 - 0.936i)T + (-0.909 - 0.415i)T^{2} \)
59 \( 1 + (-0.304 + 0.474i)T + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.254 - 0.340i)T + (-0.281 - 0.959i)T^{2} \)
71 \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.540 - 0.841i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.989 - 0.142i)T^{2} \)
89 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.142 + 0.0101i)T + (0.989 - 0.142i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.630658679086134201445113014306, −8.917064958388343021966423360067, −7.948035579923450413778259509430, −7.67517117677476174601898228380, −7.17813924350385598426020746160, −6.12802082310166430697691710416, −4.21526685617760917899060401956, −3.50675597094540825392098396560, −2.79051418649033759485558894662, −2.08847940842542297501755607174, 1.63423172127096425981967052543, 2.65566004415818061893620647560, 3.67415532419222304443835955883, 4.78165200494236665291812016619, 5.14233910523466180258194504466, 6.78385462653107604768948933974, 7.67119423284036205581095820881, 8.478182055292338418377288050761, 8.857731205458701814558458622257, 9.931245082987702554992109036788

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