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Properties

Label 2-1265-1265.1132-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.909 + 0.415i$
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  + (−0.0498 − 0.697i)3-s + (−0.540 + 0.841i)4-s + (0.989 − 0.142i)5-s + (0.506 − 0.0727i)9-s + (−0.281 − 0.959i)11-s + (0.613 + 0.334i)12-s + (−0.148 − 0.682i)15-s + (−0.415 − 0.909i)16-s + (−0.415 + 0.909i)20-s + (0.281 + 0.959i)23-s + (0.959 − 0.281i)25-s + (−0.224 − 1.03i)27-s + (1.29 + 1.49i)31-s + (−0.654 + 0.244i)33-s + (−0.212 + 0.465i)36-s + (−0.340 − 0.254i)37-s + ⋯
L(s)  = 1  + (−0.0498 − 0.697i)3-s + (−0.540 + 0.841i)4-s + (0.989 − 0.142i)5-s + (0.506 − 0.0727i)9-s + (−0.281 − 0.959i)11-s + (0.613 + 0.334i)12-s + (−0.148 − 0.682i)15-s + (−0.415 − 0.909i)16-s + (−0.415 + 0.909i)20-s + (0.281 + 0.959i)23-s + (0.959 − 0.281i)25-s + (−0.224 − 1.03i)27-s + (1.29 + 1.49i)31-s + (−0.654 + 0.244i)33-s + (−0.212 + 0.465i)36-s + (−0.340 − 0.254i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144990879\)
\(L(\frac12)\) \(\approx\) \(1.144990879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.989 + 0.142i)T \)
11 \( 1 + (0.281 + 0.959i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
good2 \( 1 + (0.540 - 0.841i)T^{2} \)
3 \( 1 + (0.0498 + 0.697i)T + (-0.989 + 0.142i)T^{2} \)
7 \( 1 + (-0.755 - 0.654i)T^{2} \)
13 \( 1 + (0.755 - 0.654i)T^{2} \)
17 \( 1 + (0.909 + 0.415i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-1.29 - 1.49i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (0.340 + 0.254i)T + (0.281 + 0.959i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.989 + 0.142i)T^{2} \)
47 \( 1 + (1.13 + 1.13i)T + iT^{2} \)
53 \( 1 + (-0.682 + 1.83i)T + (-0.755 - 0.654i)T^{2} \)
59 \( 1 + (-0.983 - 0.449i)T + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.142 + 0.989i)T^{2} \)
67 \( 1 + (1.64 - 0.898i)T + (0.540 - 0.841i)T^{2} \)
71 \( 1 + (1.84 + 0.540i)T + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.909 + 0.415i)T^{2} \)
79 \( 1 + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (0.281 + 0.959i)T^{2} \)
89 \( 1 + (1.19 - 1.37i)T + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.718 - 0.959i)T + (-0.281 + 0.959i)T^{2} \)
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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

−9.791844566880851227297057138624, −8.816895135446243702412084147168, −8.338963811974787227906333736563, −7.31978848630416960455445082037, −6.65899265127924459157999226238, −5.64422787433221871258931427071, −4.81020591369594320013462139294, −3.59728746478259739339757306815, −2.60733184312583914805467573103, −1.25540942747564719508914616496, 1.49058253885776570308989091637, 2.64569099543400950341829735140, 4.30526901495030513182259592159, 4.72170972591936716340848531099, 5.66456433707727210045337512770, 6.43690301474206739486134879796, 7.38576435778208870998655154272, 8.653180389586278200361117359872, 9.396603594252125788052618008895, 10.07712842326064643474825162814

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