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Properties

Label 2-1265-1265.527-c0-0-0
Degree $2$
Conductor $1265$
Sign $0.224 - 0.974i$
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  + (1.59 + 1.19i)3-s + (−0.909 + 0.415i)4-s + (−0.281 − 0.959i)5-s + (0.839 + 2.85i)9-s + (0.540 + 0.841i)11-s + (−1.94 − 0.424i)12-s + (0.697 − 1.86i)15-s + (0.654 − 0.755i)16-s + (0.654 + 0.755i)20-s + (−0.540 − 0.841i)23-s + (−0.841 + 0.540i)25-s + (−1.38 + 3.70i)27-s + (−0.0801 + 0.557i)31-s + (−0.142 + 1.98i)33-s + (−1.95 − 2.25i)36-s + (0.334 − 0.613i)37-s + ⋯
L(s)  = 1  + (1.59 + 1.19i)3-s + (−0.909 + 0.415i)4-s + (−0.281 − 0.959i)5-s + (0.839 + 2.85i)9-s + (0.540 + 0.841i)11-s + (−1.94 − 0.424i)12-s + (0.697 − 1.86i)15-s + (0.654 − 0.755i)16-s + (0.654 + 0.755i)20-s + (−0.540 − 0.841i)23-s + (−0.841 + 0.540i)25-s + (−1.38 + 3.70i)27-s + (−0.0801 + 0.557i)31-s + (−0.142 + 1.98i)33-s + (−1.95 − 2.25i)36-s + (0.334 − 0.613i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.425519733\)
\(L(\frac12)\) \(\approx\) \(1.425519733\)
\(L(1)\) 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 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
good2 \( 1 + (0.909 - 0.415i)T^{2} \)
3 \( 1 + (-1.59 - 1.19i)T + (0.281 + 0.959i)T^{2} \)
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 \( 1 + (-0.334 + 0.613i)T + (-0.540 - 0.841i)T^{2} \)
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 \( 1 + (-1.86 + 0.133i)T + (0.989 - 0.142i)T^{2} \)
59 \( 1 + (-1.37 + 1.19i)T + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.139 + 0.0303i)T + (0.909 - 0.415i)T^{2} \)
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 \( 1 + (-1.54 + 0.841i)T + (0.540 - 0.841i)T^{2} \)
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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

−9.783266537530965414613639017449, −9.130828750906594657330959458368, −8.592162001772716970161427781702, −8.067182572070488742863912918482, −7.22648900619675162554836783770, −5.32487834281630854132581487300, −4.55310970033929340680175964228, −4.10908636008522189450269039516, −3.29380909838122382987538441673, −1.97224138171408315703044642393, 1.21846583812730669002315519543, 2.49876678301790623389388227086, 3.49786954471185567300923313195, 4.04765097321493754433659962982, 5.84311089787348176935314157805, 6.55621650404528893965062540466, 7.43341741989812901933925104411, 8.145220303642107727523331687019, 8.737384169015889830609722660886, 9.529775154033924991922201233930

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