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Properties

Label 2-845-169.116-c1-0-58
Degree $2$
Conductor $845$
Sign $-0.984 + 0.176i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.257 + 0.291i)2-s + (0.0778 + 0.205i)3-s + (0.222 − 1.83i)4-s + (−0.239 − 0.970i)5-s + (−0.0396 + 0.0756i)6-s + (−3.51 + 2.42i)7-s + (1.23 − 0.850i)8-s + (2.20 − 1.95i)9-s + (0.220 − 0.320i)10-s + (−3.80 + 4.29i)11-s + (0.394 − 0.0971i)12-s + (−3.05 − 1.90i)13-s + (−1.61 − 0.397i)14-s + (0.180 − 0.124i)15-s + (−3.02 − 0.745i)16-s + (−2.36 − 3.42i)17-s + ⋯
L(s)  = 1  + (0.182 + 0.205i)2-s + (0.0449 + 0.118i)3-s + (0.111 − 0.917i)4-s + (−0.107 − 0.434i)5-s + (−0.0162 + 0.0308i)6-s + (−1.32 + 0.915i)7-s + (0.435 − 0.300i)8-s + (0.736 − 0.652i)9-s + (0.0698 − 0.101i)10-s + (−1.14 + 1.29i)11-s + (0.113 − 0.0280i)12-s + (−0.848 − 0.528i)13-s + (−0.430 − 0.106i)14-s + (0.0466 − 0.0322i)15-s + (−0.756 − 0.186i)16-s + (−0.573 − 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0274252 - 0.308775i\)
\(L(\frac12)\) \(\approx\) \(0.0274252 - 0.308775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.239 + 0.970i)T \)
13 \( 1 + (3.05 + 1.90i)T \)
good2 \( 1 + (-0.257 - 0.291i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.0778 - 0.205i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (3.51 - 2.42i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (3.80 - 4.29i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (2.36 + 3.42i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 + 0.348iT - 19T^{2} \)
23 \( 1 - 0.700T + 23T^{2} \)
29 \( 1 + (-1.20 + 1.06i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (-0.781 + 1.48i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (4.02 - 7.66i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (8.15 - 3.09i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (9.86 - 5.17i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (6.90 - 0.838i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (1.78 + 2.58i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (1.42 + 5.77i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (-5.95 + 8.62i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-3.52 + 0.427i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (-13.7 + 5.21i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-4.94 + 5.58i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-0.335 - 2.76i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-0.407 - 0.154i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + 2.34iT - 89T^{2} \)
97 \( 1 + (0.108 - 0.441i)T + (-85.8 - 45.0i)T^{2} \)
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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

−9.842631608311851150059074338940, −9.358278764383400808055772230059, −8.058495639723510224664899832729, −6.85778787686073099961637159605, −6.49388177222353548350270522731, −5.02997241730059217209734497263, −4.90245588570221403734894127662, −3.17833942013084162435434512627, −2.04864923228569002359988992426, −0.12805780584459935977576129835, 2.25245635090979215527798230934, 3.28858168407452082280369566540, 3.96424563914315226573543087781, 5.16933074301915378340287953110, 6.63887016724948901284805988117, 7.08544823294937940064202786400, 7.945816223752278902854765150957, 8.756135769218312455914657040744, 10.10228405504293614136091842188, 10.50485478813154476310265260587

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