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Properties

Label 2-1265-1265.362-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.735 + 0.677i$
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  + (1.40 − 0.767i)3-s + (0.755 − 0.654i)4-s + (−0.540 + 0.841i)5-s + (0.845 − 1.31i)9-s + (−0.909 + 0.415i)11-s + (0.559 − 1.50i)12-s + (−0.114 + 1.59i)15-s + (0.142 − 0.989i)16-s + (0.142 + 0.989i)20-s + (0.909 − 0.415i)23-s + (−0.415 − 0.909i)25-s + (0.0643 − 0.899i)27-s + (−1.03 + 0.304i)31-s + (−0.959 + 1.28i)33-s + (−0.222 − 1.54i)36-s + (0.139 − 0.0303i)37-s + ⋯
L(s)  = 1  + (1.40 − 0.767i)3-s + (0.755 − 0.654i)4-s + (−0.540 + 0.841i)5-s + (0.845 − 1.31i)9-s + (−0.909 + 0.415i)11-s + (0.559 − 1.50i)12-s + (−0.114 + 1.59i)15-s + (0.142 − 0.989i)16-s + (0.142 + 0.989i)20-s + (0.909 − 0.415i)23-s + (−0.415 − 0.909i)25-s + (0.0643 − 0.899i)27-s + (−1.03 + 0.304i)31-s + (−0.959 + 1.28i)33-s + (−0.222 − 1.54i)36-s + (0.139 − 0.0303i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.735 + 0.677i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.727949659\)
\(L(\frac12)\) \(\approx\) \(1.727949659\)
\(L(1)\) 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 \( 1 + (0.909 - 0.415i)T \)
23 \( 1 + (-0.909 + 0.415i)T \)
good2 \( 1 + (-0.755 + 0.654i)T^{2} \)
3 \( 1 + (-1.40 + 0.767i)T + (0.540 - 0.841i)T^{2} \)
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 \( 1 + (-0.139 + 0.0303i)T + (0.909 - 0.415i)T^{2} \)
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 \( 1 + (1.59 - 1.19i)T + (0.281 - 0.959i)T^{2} \)
59 \( 1 + (1.49 - 0.215i)T + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.418 + 1.12i)T + (-0.755 + 0.654i)T^{2} \)
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 \( 1 + (-0.0903 + 0.415i)T + (-0.909 - 0.415i)T^{2} \)
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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

−9.697012857351452300790921209241, −8.936811008224168114582022570878, −7.78933986081250862965946950640, −7.50211432336972716702229604273, −6.79369145180683958323820427128, −5.87918022828315744069188342979, −4.51202759991384848290328675707, −3.06856888077579484302370231113, −2.70481198030886341716617305003, −1.60499862513657009616035627716, 1.97339262219989530286438261261, 3.07719502418126464927862566655, 3.64155956655878019116873576351, 4.61111929964508302421134834528, 5.60727276361543943792909833318, 7.11616329305133331125007751480, 7.78253620434492254337263153946, 8.401201191357837015819306924964, 8.941894089842223484245334394908, 9.786831858561046862952057397744

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