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Properties

Label 2-1265-1265.318-c0-0-0
Degree $2$
Conductor $1265$
Sign $0.999 - 0.0309i$
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  + (−1.05 − 0.574i)3-s + (0.755 + 0.654i)4-s + (−0.989 − 0.142i)5-s + (0.236 + 0.367i)9-s + (0.909 + 0.415i)11-s + (−0.418 − 1.12i)12-s + (0.959 + 0.718i)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 + (0.0481 + 0.673i)27-s + (1.03 + 0.304i)31-s + (−0.718 − 0.959i)33-s + (−0.0621 + 0.432i)36-s + (1.94 + 0.424i)37-s + ⋯
L(s)  = 1  + (−1.05 − 0.574i)3-s + (0.755 + 0.654i)4-s + (−0.989 − 0.142i)5-s + (0.236 + 0.367i)9-s + (0.909 + 0.415i)11-s + (−0.418 − 1.12i)12-s + (0.959 + 0.718i)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 + (0.0481 + 0.673i)27-s + (1.03 + 0.304i)31-s + (−0.718 − 0.959i)33-s + (−0.0621 + 0.432i)36-s + (1.94 + 0.424i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.999 - 0.0309i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.999 - 0.0309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7904533803\)
\(L(\frac12)\) \(\approx\) \(0.7904533803\)
\(L(1)\) 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 \( 1 + (1.05 + 0.574i)T + (0.540 + 0.841i)T^{2} \)
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 \( 1 + (-1.94 - 0.424i)T + (0.909 + 0.415i)T^{2} \)
41 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.540 + 0.841i)T^{2} \)
47 \( 1 + (-0.300 + 0.300i)T - iT^{2} \)
53 \( 1 + (0.114 + 0.0855i)T + (0.281 + 0.959i)T^{2} \)
59 \( 1 + (-1.49 - 0.215i)T + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (0.559 - 1.50i)T + (-0.755 - 0.654i)T^{2} \)
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 \( 1 + (0.415 + 1.90i)T + (-0.909 + 0.415i)T^{2} \)
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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

−10.09230520162922771318483239814, −8.823255604293925864004918479796, −8.128013904945295964969126444939, −7.16680347159992410444604893251, −6.75331723497991028892767552181, −5.98904730905513474238759469657, −4.71419100150097392970474596886, −3.86589520960124911914328333281, −2.70365880048636550336930731665, −1.14203223939616845771320464952, 1.01485944148036493152162150235, 2.77736889462807821686391569124, 3.98880587603132990334431426347, 4.81187474034080101209522142343, 5.80747086272788473572929796518, 6.38253061345170604931355954357, 7.27250146878428264308250400187, 8.133545089142141268583358386574, 9.318039025482008238954939213141, 10.04327190017447266423805936539

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