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Properties

Label 2-1265-1265.868-c0-0-0
Degree $2$
Conductor $1265$
Sign $-0.419 - 0.907i$
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  + (0.767 + 1.40i)3-s + (−0.755 + 0.654i)4-s + (0.989 − 0.142i)5-s + (−0.845 + 1.31i)9-s + (−0.909 + 0.415i)11-s + (−1.50 − 0.559i)12-s + (0.959 + 1.28i)15-s + (0.142 − 0.989i)16-s + (−0.654 + 0.755i)20-s + (0.415 + 0.909i)23-s + (0.959 − 0.281i)25-s + (−0.899 − 0.0643i)27-s + (−1.03 + 0.304i)31-s + (−1.28 − 0.959i)33-s + (−0.222 − 1.54i)36-s + (−0.0303 − 0.139i)37-s + ⋯
L(s)  = 1  + (0.767 + 1.40i)3-s + (−0.755 + 0.654i)4-s + (0.989 − 0.142i)5-s + (−0.845 + 1.31i)9-s + (−0.909 + 0.415i)11-s + (−1.50 − 0.559i)12-s + (0.959 + 1.28i)15-s + (0.142 − 0.989i)16-s + (−0.654 + 0.755i)20-s + (0.415 + 0.909i)23-s + (0.959 − 0.281i)25-s + (−0.899 − 0.0643i)27-s + (−1.03 + 0.304i)31-s + (−1.28 − 0.959i)33-s + (−0.222 − 1.54i)36-s + (−0.0303 − 0.139i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ -0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268427877\)
\(L(\frac12)\) \(\approx\) \(1.268427877\)
\(L(1)\) 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 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good2 \( 1 + (0.755 - 0.654i)T^{2} \)
3 \( 1 + (-0.767 - 1.40i)T + (-0.540 + 0.841i)T^{2} \)
7 \( 1 + (-0.281 + 0.959i)T^{2} \)
13 \( 1 + (0.281 + 0.959i)T^{2} \)
17 \( 1 + (-0.989 + 0.142i)T^{2} \)
19 \( 1 + (0.142 - 0.989i)T^{2} \)
29 \( 1 + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (1.03 - 0.304i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (0.0303 + 0.139i)T + (-0.909 + 0.415i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.540 + 0.841i)T^{2} \)
47 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
53 \( 1 + (1.19 + 1.59i)T + (-0.281 + 0.959i)T^{2} \)
59 \( 1 + (-1.49 + 0.215i)T + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (1.12 - 0.418i)T + (0.755 - 0.654i)T^{2} \)
71 \( 1 + (0.345 - 0.755i)T + (-0.654 - 0.755i)T^{2} \)
73 \( 1 + (0.989 + 0.142i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (-0.909 + 0.415i)T^{2} \)
89 \( 1 + (-1.89 - 0.557i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.415 + 0.0903i)T + (0.909 + 0.415i)T^{2} \)
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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

−10.04901988981305406958212276039, −9.252115352304045172259459306291, −8.838392890909257603274451878383, −8.016425376616215534532885331664, −7.04460598325929799503096884651, −5.38293801382558773170482677047, −5.10932176136391179417819116311, −4.03355807683886157686407507116, −3.25007116108599210327641513817, −2.25247644711520867819422995478, 1.09584188699453839423952550851, 2.18882456922286416356834774633, 3.04265019206759874176656917997, 4.60819036375395139522886524956, 5.71112823640616785093888148834, 6.21157365145701062482264098761, 7.23490459980352212559162139326, 8.012803788081633466759230687386, 8.906756849971993769831074490797, 9.306444187348712244565980103394

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