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Properties

Label 2-1265-1265.43-c0-0-0
Degree $2$
Conductor $1265$
Sign $-0.0313 - 0.999i$
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  + (0.398 − 0.148i)3-s + (0.281 + 0.959i)4-s + (−0.755 + 0.654i)5-s + (−0.619 + 0.536i)9-s + (0.989 + 0.142i)11-s + (0.254 + 0.340i)12-s + (−0.203 + 0.373i)15-s + (−0.841 + 0.540i)16-s + (−0.841 − 0.540i)20-s + (−0.989 − 0.142i)23-s + (0.142 − 0.989i)25-s + (−0.370 + 0.678i)27-s + (0.627 + 1.37i)31-s + (0.415 − 0.0903i)33-s + (−0.689 − 0.442i)36-s + (−0.0683 + 0.956i)37-s + ⋯
L(s)  = 1  + (0.398 − 0.148i)3-s + (0.281 + 0.959i)4-s + (−0.755 + 0.654i)5-s + (−0.619 + 0.536i)9-s + (0.989 + 0.142i)11-s + (0.254 + 0.340i)12-s + (−0.203 + 0.373i)15-s + (−0.841 + 0.540i)16-s + (−0.841 − 0.540i)20-s + (−0.989 − 0.142i)23-s + (0.142 − 0.989i)25-s + (−0.370 + 0.678i)27-s + (0.627 + 1.37i)31-s + (0.415 − 0.0903i)33-s + (−0.689 − 0.442i)36-s + (−0.0683 + 0.956i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0313 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0313 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1265\)    =    \(5 \cdot 11 \cdot 23\)
Sign: $-0.0313 - 0.999i$
Analytic conductor: \(0.631317\)
Root analytic conductor: \(0.794554\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1265} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1265,\ (\ :0),\ -0.0313 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052109416\)
\(L(\frac12)\) \(\approx\) \(1.052109416\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.755 - 0.654i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
23 \( 1 + (0.989 + 0.142i)T \)
good2 \( 1 + (-0.281 - 0.959i)T^{2} \)
3 \( 1 + (-0.398 + 0.148i)T + (0.755 - 0.654i)T^{2} \)
7 \( 1 + (-0.909 - 0.415i)T^{2} \)
13 \( 1 + (0.909 - 0.415i)T^{2} \)
17 \( 1 + (0.540 - 0.841i)T^{2} \)
19 \( 1 + (-0.841 + 0.540i)T^{2} \)
29 \( 1 + (0.841 + 0.540i)T^{2} \)
31 \( 1 + (-0.627 - 1.37i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 + (0.0683 - 0.956i)T + (-0.989 - 0.142i)T^{2} \)
41 \( 1 + (0.142 + 0.989i)T^{2} \)
43 \( 1 + (0.755 - 0.654i)T^{2} \)
47 \( 1 + (-0.100 + 0.100i)T - iT^{2} \)
53 \( 1 + (-0.373 + 1.71i)T + (-0.909 - 0.415i)T^{2} \)
59 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + (-1.17 + 1.56i)T + (-0.281 - 0.959i)T^{2} \)
71 \( 1 + (0.0405 + 0.281i)T + (-0.959 + 0.281i)T^{2} \)
73 \( 1 + (-0.540 - 0.841i)T^{2} \)
79 \( 1 + (-0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.989 - 0.142i)T^{2} \)
89 \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-1.98 + 0.142i)T + (0.989 - 0.142i)T^{2} \)
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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

−10.18632613702627358348347863203, −8.968058304755299573965166828631, −8.337151069524412633043885141126, −7.74266545950883443426374954863, −6.93064762313582797981578397124, −6.26543607275295445134550398437, −4.76920402805256940425002195766, −3.76259841246064684091951897562, −3.10648481247039723930018275556, −2.08394210292917337806291909073, 0.884265203369929879497927880609, 2.31873928207491667087945879164, 3.70535587934279940155435288525, 4.38031138061005216290885690035, 5.60362024072072822485138045363, 6.20398701481160461325696709901, 7.24508233556870509823441852076, 8.183931614061830754131119011018, 9.021344193542027291870243765372, 9.467752104286506201721335953676

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