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Properties

Label 2-845-65.7-c1-0-12
Degree $2$
Conductor $845$
Sign $0.991 + 0.126i$
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  + (−0.866 − 0.5i)2-s + (0.366 + 1.36i)3-s + (−0.500 − 0.866i)4-s + (−2 − i)5-s + (0.366 − 1.36i)6-s + (1 + 1.73i)7-s + 3i·8-s + (0.866 − 0.5i)9-s + (1.23 + 1.86i)10-s + (−0.366 − 1.36i)11-s + (0.999 − i)12-s − 1.99i·14-s + (0.633 − 3.09i)15-s + (0.500 − 0.866i)16-s + (1.36 + 0.366i)17-s − 18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.211 + 0.788i)3-s + (−0.250 − 0.433i)4-s + (−0.894 − 0.447i)5-s + (0.149 − 0.557i)6-s + (0.377 + 0.654i)7-s + 1.06i·8-s + (0.288 − 0.166i)9-s + (0.389 + 0.590i)10-s + (−0.110 − 0.411i)11-s + (0.288 − 0.288i)12-s − 0.534i·14-s + (0.163 − 0.799i)15-s + (0.125 − 0.216i)16-s + (0.331 + 0.0887i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.956356 - 0.0607369i\)
\(L(\frac12)\) \(\approx\) \(0.956356 - 0.0607369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.366 - 1.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.366 + 1.36i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.36 - 0.366i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.83 + 1.83i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.09 + 1.09i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 - 5i)T + 31iT^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.56 + 2.56i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.366 + 1.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-2.56 + 9.56i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 - 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.366 + 1.36i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-6.83 + 1.83i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24343923532819926644613250141, −9.130651528527073695652168160288, −8.759648041894502615537226403542, −8.127769123656072130740495433498, −6.82012336302151151092570130347, −5.48985539108135282744121919558, −4.72891646305444471610451719586, −3.90408835595599430050728113651, −2.50209419819289814601552648467, −0.893210652838899622041517008259, 0.868612938054539789621752417576, 2.54272976493798256789409814973, 3.94994644912632003462453801905, 4.52825733532712265312220644415, 6.38065377938440655514055713831, 7.12338087571802350040542039832, 7.75589007786310678918162149006, 8.113202537448552600784157475710, 9.132621401544678723312042178425, 10.24117846924266228253143435837

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