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Properties

Label 2-845-13.12-c1-0-40
Degree $2$
Conductor $845$
Sign $0.0304 + 0.999i$
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  + 0.271i·2-s − 0.319·3-s + 1.92·4-s − i·5-s − 0.0867i·6-s − 3.38i·7-s + 1.06i·8-s − 2.89·9-s + 0.271·10-s − 1.75i·11-s − 0.615·12-s + 0.917·14-s + 0.319i·15-s + 3.56·16-s − 1.95·17-s − 0.786i·18-s + ⋯
L(s)  = 1  + 0.191i·2-s − 0.184·3-s + 0.963·4-s − 0.447i·5-s − 0.0354i·6-s − 1.27i·7-s + 0.376i·8-s − 0.965·9-s + 0.0858·10-s − 0.528i·11-s − 0.177·12-s + 0.245·14-s + 0.0825i·15-s + 0.890·16-s − 0.473·17-s − 0.185i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0304 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0304 + 0.999i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0304 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02779 - 0.996929i\)
\(L(\frac12)\) \(\approx\) \(1.02779 - 0.996929i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.271iT - 2T^{2} \)
3 \( 1 + 0.319T + 3T^{2} \)
7 \( 1 + 3.38iT - 7T^{2} \)
11 \( 1 + 1.75iT - 11T^{2} \)
17 \( 1 + 1.95T + 17T^{2} \)
19 \( 1 + 7.13iT - 19T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 + 10.5iT - 37T^{2} \)
41 \( 1 - 0.911iT - 41T^{2} \)
43 \( 1 + 4.58T + 43T^{2} \)
47 \( 1 - 8.58iT - 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 3.82iT - 59T^{2} \)
61 \( 1 - 7.98T + 61T^{2} \)
67 \( 1 - 0.472iT - 67T^{2} \)
71 \( 1 + 5.50iT - 71T^{2} \)
73 \( 1 + 2.93iT - 73T^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 3.85iT - 89T^{2} \)
97 \( 1 + 6.95iT - 97T^{2} \)
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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

−10.24045938291718214024946077469, −9.003070188892429008332774029525, −8.210793213438598996405457416603, −7.33280047133422594812378342386, −6.55973782043030958126318447781, −5.73852755824858119877787659591, −4.66726372379214083974466866960, −3.48711040799218681781546384016, −2.30369687894783428480446189376, −0.66157579030691434203374443346, 1.97021588517625936823118546930, 2.66090431641148807609742710695, 3.81915299416774342564947679393, 5.43720000321315204351255894225, 6.04386623381263465127563915434, 6.75336819460804864644737767705, 7.995379266787846482664459194049, 8.528366694973383923483447961150, 9.874945048164385088060088090141, 10.31605512135751073087347608012

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