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Properties

Label 2-845-65.8-c1-0-53
Degree $2$
Conductor $845$
Sign $0.850 - 0.525i$
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  + 2.25·2-s + (1.40 + 1.40i)3-s + 3.06·4-s + (2.22 − 0.247i)5-s + (3.16 + 3.16i)6-s − 1.27i·7-s + 2.39·8-s + 0.947i·9-s + (5.00 − 0.558i)10-s + (−3.86 + 3.86i)11-s + (4.30 + 4.30i)12-s − 2.87i·14-s + (3.47 + 2.77i)15-s − 0.731·16-s + (−2.27 − 2.27i)17-s + 2.13i·18-s + ⋯
L(s)  = 1  + 1.59·2-s + (0.811 + 0.811i)3-s + 1.53·4-s + (0.993 − 0.110i)5-s + (1.29 + 1.29i)6-s − 0.482i·7-s + 0.848·8-s + 0.315i·9-s + (1.58 − 0.176i)10-s + (−1.16 + 1.16i)11-s + (1.24 + 1.24i)12-s − 0.768i·14-s + (0.896 + 0.716i)15-s − 0.182·16-s + (−0.552 − 0.552i)17-s + 0.502i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.85675 + 1.37897i\)
\(L(\frac12)\) \(\approx\) \(4.85675 + 1.37897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.247i)T \)
13 \( 1 \)
good2 \( 1 - 2.25T + 2T^{2} \)
3 \( 1 + (-1.40 - 1.40i)T + 3iT^{2} \)
7 \( 1 + 1.27iT - 7T^{2} \)
11 \( 1 + (3.86 - 3.86i)T - 11iT^{2} \)
17 \( 1 + (2.27 + 2.27i)T + 17iT^{2} \)
19 \( 1 + (0.861 - 0.861i)T - 19iT^{2} \)
23 \( 1 + (0.117 - 0.117i)T - 23iT^{2} \)
29 \( 1 + 9.71iT - 29T^{2} \)
31 \( 1 + (0.233 + 0.233i)T + 31iT^{2} \)
37 \( 1 - 1.32iT - 37T^{2} \)
41 \( 1 + (0.354 + 0.354i)T + 41iT^{2} \)
43 \( 1 + (4.71 - 4.71i)T - 43iT^{2} \)
47 \( 1 - 3.20iT - 47T^{2} \)
53 \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \)
59 \( 1 + (0.00162 + 0.00162i)T + 59iT^{2} \)
61 \( 1 - 1.39T + 61T^{2} \)
67 \( 1 + 6.07T + 67T^{2} \)
71 \( 1 + (-8.59 - 8.59i)T + 71iT^{2} \)
73 \( 1 + 7.34T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 2.65iT - 83T^{2} \)
89 \( 1 + (5.09 + 5.09i)T + 89iT^{2} \)
97 \( 1 + 4.18T + 97T^{2} \)
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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

−10.09386203414184838559710947489, −9.751576734063844994376369860589, −8.688725944434454492609218842837, −7.49883017545754205964287201003, −6.53617345618148159831177334179, −5.57339686637105183400720852171, −4.65878512720324640418330720180, −4.16335777423164040286339520353, −2.87217741399890115198479326673, −2.24317863910216804516249620147, 1.90986434916714776985305594773, 2.67958988363961718872082941457, 3.41820292338283899227427207106, 5.00485317517287691905619003908, 5.58565706537976617875465328696, 6.44417684109532440420653566500, 7.24172747234355714582943833817, 8.476807727515416232781568108568, 8.946180391255686259141941381906, 10.46901125553644420148490230010

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