Properties

Label 2-845-13.12-c1-0-19
Degree $2$
Conductor $845$
Sign $-0.960 + 0.277i$
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  − 2.49i·2-s − 2.82·3-s − 4.22·4-s + i·5-s + 7.05i·6-s + 1.90i·7-s + 5.55i·8-s + 4.99·9-s + 2.49·10-s + 1.06i·11-s + 11.9·12-s + 4.75·14-s − 2.82i·15-s + 5.41·16-s − 0.637·17-s − 12.4i·18-s + ⋯
L(s)  = 1  − 1.76i·2-s − 1.63·3-s − 2.11·4-s + 0.447i·5-s + 2.87i·6-s + 0.720i·7-s + 1.96i·8-s + 1.66·9-s + 0.789·10-s + 0.322i·11-s + 3.44·12-s + 1.27·14-s − 0.729i·15-s + 1.35·16-s − 0.154·17-s − 2.93i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.960 + 0.277i$
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: \((2,\ 845,\ (\ :1/2),\ -0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0763579 - 0.539824i\)
\(L(\frac12)\) \(\approx\) \(0.0763579 - 0.539824i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.49iT - 2T^{2} \)
3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 - 1.90iT - 7T^{2} \)
11 \( 1 - 1.06iT - 11T^{2} \)
17 \( 1 + 0.637T + 17T^{2} \)
19 \( 1 + 5.73iT - 19T^{2} \)
23 \( 1 + 3.81T + 23T^{2} \)
29 \( 1 - 9.45T + 29T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 - 0.757iT - 37T^{2} \)
41 \( 1 + 0.267iT - 41T^{2} \)
43 \( 1 + 0.637T + 43T^{2} \)
47 \( 1 + 9.44iT - 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 - 0.741iT - 59T^{2} \)
61 \( 1 - 4.19T + 61T^{2} \)
67 \( 1 + 8.09iT - 67T^{2} \)
71 \( 1 + 9.76iT - 71T^{2} \)
73 \( 1 + 3.71iT - 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 5.11iT - 83T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 - 4.22iT - 97T^{2} \)
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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

−10.29863352910179818346909875077, −9.446254388074137125247459243824, −8.497742639463419770356293913513, −6.98844157765042815479456754122, −6.10557577569730479661190086056, −5.03373832919581499192822931950, −4.43891496884894719143400793679, −3.05854680870113004097050015692, −1.94379476364255415255890412747, −0.45988184234573383236156157737, 0.934345043060737349453222619571, 4.05487596131246268194991160110, 4.71060046514453868883198367887, 5.64341401542063775161391613088, 6.16281244814765631426876172867, 6.89857457924155547949880330183, 7.79607426144715470351554157393, 8.515537175935628676142470675100, 9.766445366955942797492194118095, 10.38127239498312013657664468299

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