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Properties

Label 2-1265-1265.1187-c0-0-1
Degree $2$
Conductor $1265$
Sign $0.859 - 0.510i$
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  + (0.697 + 0.0498i)3-s + (0.540 + 0.841i)4-s + (0.909 − 0.415i)5-s + (−0.506 − 0.0727i)9-s + (−0.281 + 0.959i)11-s + (0.334 + 0.613i)12-s + (0.654 − 0.244i)15-s + (−0.415 + 0.909i)16-s + (0.841 + 0.540i)20-s + (−0.959 − 0.281i)23-s + (0.654 − 0.755i)25-s + (−1.03 − 0.224i)27-s + (1.29 − 1.49i)31-s + (−0.244 + 0.654i)33-s + (−0.212 − 0.465i)36-s + (−0.254 − 0.340i)37-s + ⋯
L(s)  = 1  + (0.697 + 0.0498i)3-s + (0.540 + 0.841i)4-s + (0.909 − 0.415i)5-s + (−0.506 − 0.0727i)9-s + (−0.281 + 0.959i)11-s + (0.334 + 0.613i)12-s + (0.654 − 0.244i)15-s + (−0.415 + 0.909i)16-s + (0.841 + 0.540i)20-s + (−0.959 − 0.281i)23-s + (0.654 − 0.755i)25-s + (−1.03 − 0.224i)27-s + (1.29 − 1.49i)31-s + (−0.244 + 0.654i)33-s + (−0.212 − 0.465i)36-s + (−0.254 − 0.340i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $0.859 - 0.510i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (1187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ 0.859 - 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.597224712\)
\(L(\frac12)\) \(\approx\) \(1.597224712\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.909 + 0.415i)T \)
11 \( 1 + (0.281 - 0.959i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
good2 \( 1 + (-0.540 - 0.841i)T^{2} \)
3 \( 1 + (-0.697 - 0.0498i)T + (0.989 + 0.142i)T^{2} \)
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 \( 1 + (0.254 + 0.340i)T + (-0.281 + 0.959i)T^{2} \)
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 \( 1 + (-1.83 + 0.682i)T + (0.755 - 0.654i)T^{2} \)
59 \( 1 + (0.983 - 0.449i)T + (0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.142 - 0.989i)T^{2} \)
67 \( 1 + (-0.898 + 1.64i)T + (-0.540 - 0.841i)T^{2} \)
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 \( 1 + (-0.959 - 0.718i)T + (0.281 + 0.959i)T^{2} \)
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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

−9.816334601771413389201291964036, −9.100034260728948618632736998333, −8.252093492132171967843827538948, −7.75714574066864990957611106332, −6.66144618669134800104305608663, −5.91134219646755074947254624986, −4.75023937964176892964223177964, −3.73850475820549422705924472122, −2.56909953496599690162862273720, −2.02920205789883530401391634685, 1.53605328183081804304882035269, 2.60945962707786871647712798905, 3.26145723510240538333511707240, 4.93375961126421729807628979412, 5.83737943779774958139700182551, 6.30395723936743190801374039506, 7.30734346097913809910913283665, 8.337256636501923076141225672626, 8.999224276495988300768960916700, 9.961630366246246141488924294190

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