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Properties

Label 2-845-13.4-c1-0-42
Degree $2$
Conductor $845$
Sign $0.967 - 0.252i$
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  + (2.16 + 1.24i)2-s + (1.41 − 2.44i)3-s + (2.11 + 3.66i)4-s + i·5-s + (6.10 − 3.52i)6-s + (1.64 − 0.952i)7-s + 5.55i·8-s + (−2.49 − 4.32i)9-s + (−1.24 + 2.16i)10-s + (−0.926 − 0.534i)11-s + 11.9·12-s + 4.75·14-s + (2.44 + 1.41i)15-s + (−2.70 + 4.69i)16-s + (0.318 + 0.551i)17-s − 12.4i·18-s + ⋯
L(s)  = 1  + (1.52 + 0.882i)2-s + (0.816 − 1.41i)3-s + (1.05 + 1.83i)4-s + 0.447i·5-s + (2.49 − 1.43i)6-s + (0.623 − 0.360i)7-s + 1.96i·8-s + (−0.831 − 1.44i)9-s + (−0.394 + 0.683i)10-s + (−0.279 − 0.161i)11-s + 3.44·12-s + 1.27·14-s + (0.632 + 0.364i)15-s + (−0.677 + 1.17i)16-s + (0.0772 + 0.133i)17-s − 2.93i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.967 - 0.252i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.67025 + 0.599625i\)
\(L(\frac12)\) \(\approx\) \(4.67025 + 0.599625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-2.16 - 1.24i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.41 + 2.44i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.64 + 0.952i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.926 + 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.318 - 0.551i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.90 + 3.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.72 - 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (0.655 + 0.378i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.232 - 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.318 - 0.551i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + (-0.641 + 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.01 - 4.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.45 - 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.71iT - 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11iT - 83T^{2} \)
89 \( 1 + (-10.8 - 6.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.65 + 2.11i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.50457478493230518183549707596, −8.780976615044642887089747056400, −8.081782182794717170288951525597, −7.40189274767835770017328099724, −6.78694237607060054421485902147, −6.07474020248533599285294455134, −4.98539348097294821447957683404, −3.81303934003622706954347466096, −2.90852282956867254796556878153, −1.81023599639487911648198610643, 1.98740678941239872205744928435, 2.87642520884560957235645193864, 3.91131876622579047076076689004, 4.57469762509582880872991030435, 5.15391193544780901055499744568, 6.10638173614328839718757741655, 7.76990870134379619565013455679, 8.773308352090656109912569931691, 9.553206969779963737859503061412, 10.32118273033138095247282438058

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