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Properties

Label 2-1265-1265.1033-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.971 - 0.235i$
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  + (0.0855 − 0.114i)3-s + (0.909 − 0.415i)4-s + (0.281 + 0.959i)5-s + (0.275 + 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.0303 − 0.139i)12-s + (0.133 + 0.0498i)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 + (0.264 + 0.0987i)27-s + (0.0801 − 0.557i)31-s + (−0.142 − 0.0101i)33-s + (0.641 + 0.740i)36-s + (−1.64 − 0.898i)37-s + ⋯
L(s)  = 1  + (0.0855 − 0.114i)3-s + (0.909 − 0.415i)4-s + (0.281 + 0.959i)5-s + (0.275 + 0.939i)9-s + (−0.540 − 0.841i)11-s + (0.0303 − 0.139i)12-s + (0.133 + 0.0498i)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 + (0.264 + 0.0987i)27-s + (0.0801 − 0.557i)31-s + (−0.142 − 0.0101i)33-s + (0.641 + 0.740i)36-s + (−1.64 − 0.898i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.971 - 0.235i$
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: \((2,\ 1265,\ (\ :0),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396738983\)
\(L(\frac12)\) \(\approx\) \(1.396738983\)
\(L(1)\) 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 \( 1 + (-0.0855 + 0.114i)T + (-0.281 - 0.959i)T^{2} \)
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 \( 1 + (1.64 + 0.898i)T + (0.540 + 0.841i)T^{2} \)
41 \( 1 + (-0.841 - 0.540i)T^{2} \)
43 \( 1 + (-0.281 - 0.959i)T^{2} \)
47 \( 1 + (1.24 - 1.24i)T - iT^{2} \)
53 \( 1 + (-0.0498 - 0.697i)T + (-0.989 + 0.142i)T^{2} \)
59 \( 1 + (-1.37 + 1.19i)T + (0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.959 - 0.281i)T^{2} \)
67 \( 1 + (0.424 + 1.94i)T + (-0.909 + 0.415i)T^{2} \)
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 \( 1 + (-0.459 - 0.841i)T + (-0.540 + 0.841i)T^{2} \)
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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

−10.17261108811510250783202672511, −9.241055548568046225001870327951, −7.986204440304503947936210718824, −7.43597039665272044772473809035, −6.65608018448378271508571104504, −5.80403771250292164751693902087, −5.10998088538808610076884403074, −3.49532563703754955595946903271, −2.64065161807801621315818126392, −1.72759748527625001754975551511, 1.44266706415850949693418859757, 2.59337352436776676686838900605, 3.73403336225478542628790020832, 4.73007478006125247187280203765, 5.66066279917166376545594064950, 6.73450660368683529891165980019, 7.23252427667308118362281668466, 8.427653227721978284601792569008, 8.836408681816996864240864387138, 10.06778448791200326858470477725

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