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Properties

Label 2-1265-1265.1033-c0-0-0
Degree $2$
Conductor $1265$
Sign $0.786 - 0.617i$
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  + (−1.19 + 1.59i)3-s + (0.909 − 0.415i)4-s + (0.755 − 0.654i)5-s + (−0.839 − 2.85i)9-s + (0.540 + 0.841i)11-s + (−0.424 + 1.94i)12-s + (0.142 + 1.98i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (3.70 + 1.38i)27-s + (−0.0801 + 0.557i)31-s + (−1.98 − 0.142i)33-s + (−1.95 − 2.25i)36-s + (−0.613 − 0.334i)37-s + ⋯
L(s)  = 1  + (−1.19 + 1.59i)3-s + (0.909 − 0.415i)4-s + (0.755 − 0.654i)5-s + (−0.839 − 2.85i)9-s + (0.540 + 0.841i)11-s + (−0.424 + 1.94i)12-s + (0.142 + 1.98i)15-s + (0.654 − 0.755i)16-s + (0.415 − 0.909i)20-s + (0.841 − 0.540i)23-s + (0.142 − 0.989i)25-s + (3.70 + 1.38i)27-s + (−0.0801 + 0.557i)31-s + (−1.98 − 0.142i)33-s + (−1.95 − 2.25i)36-s + (−0.613 − 0.334i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.786 - 0.617i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.786 - 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.040954481\)
\(L(\frac12)\) \(\approx\) \(1.040954481\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (-0.841 + 0.540i)T \)
good2 \( 1 + (-0.909 + 0.415i)T^{2} \)
3 \( 1 + (1.19 - 1.59i)T + (-0.281 - 0.959i)T^{2} \)
7 \( 1 + (-0.989 + 0.142i)T^{2} \)
13 \( 1 + (0.989 + 0.142i)T^{2} \)
17 \( 1 + (-0.755 + 0.654i)T^{2} \)
19 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (-0.654 - 0.755i)T^{2} \)
31 \( 1 + (0.0801 - 0.557i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + (0.613 + 0.334i)T + (0.540 + 0.841i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.281 - 0.959i)T^{2} \)
47 \( 1 + (0.677 - 0.677i)T - iT^{2} \)
53 \( 1 + (-0.133 - 1.86i)T + (-0.989 + 0.142i)T^{2} \)
59 \( 1 + (1.37 - 1.19i)T + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.0303 + 0.139i)T + (-0.909 + 0.415i)T^{2} \)
71 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.755 + 0.654i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (0.540 + 0.841i)T^{2} \)
89 \( 1 + (-0.215 - 1.49i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.841 + 1.54i)T + (-0.540 + 0.841i)T^{2} \)
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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

−10.16866512519490247876204344137, −9.347955408839502798402343754321, −8.881775038103007081060595643899, −7.15925275587196410678358600817, −6.31755472132486520231747133964, −5.73278697977155677900969372696, −4.92862607456369035568042861897, −4.29065407337723314537780448443, −2.93743756239992938636831116954, −1.30810041960204307561733875000, 1.36613052924544473530824701388, 2.24746010450373425300338363097, 3.28096072206190159818033489725, 5.21962878955182393272027341628, 5.95863086240151242770647331641, 6.54363347591851509383399296285, 7.07162081476132566138663092124, 7.79582228198460565688661438241, 8.728041409955998208109413960570, 10.16348037509263100579878534189

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