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Properties

Label 2-1160-145.17-c1-0-24
Degree $2$
Conductor $1160$
Sign $0.428 - 0.903i$
Analytic cond. $9.26264$
Root an. cond. $3.04345$
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  + 1.84i·3-s + (2.19 + 0.412i)5-s + (0.563 + 0.563i)7-s − 0.409·9-s + (2.03 − 2.03i)11-s + (−0.859 − 0.859i)13-s + (−0.761 + 4.05i)15-s + 2.05·17-s + (2.99 + 2.99i)19-s + (−1.04 + 1.04i)21-s + (4.58 − 4.58i)23-s + (4.66 + 1.81i)25-s + 4.78i·27-s + (3.13 − 4.37i)29-s + (−6.55 + 6.55i)31-s + ⋯
L(s)  = 1  + 1.06i·3-s + (0.982 + 0.184i)5-s + (0.212 + 0.212i)7-s − 0.136·9-s + (0.614 − 0.614i)11-s + (−0.238 − 0.238i)13-s + (−0.196 + 1.04i)15-s + 0.497·17-s + (0.686 + 0.686i)19-s + (−0.226 + 0.226i)21-s + (0.955 − 0.955i)23-s + (0.932 + 0.362i)25-s + 0.920i·27-s + (0.581 − 0.813i)29-s + (−1.17 + 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $0.428 - 0.903i$
Analytic conductor: \(9.26264\)
Root analytic conductor: \(3.04345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1160} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1160,\ (\ :1/2),\ 0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.200477709\)
\(L(\frac12)\) \(\approx\) \(2.200477709\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.19 - 0.412i)T \)
29 \( 1 + (-3.13 + 4.37i)T \)
good3 \( 1 - 1.84iT - 3T^{2} \)
7 \( 1 + (-0.563 - 0.563i)T + 7iT^{2} \)
11 \( 1 + (-2.03 + 2.03i)T - 11iT^{2} \)
13 \( 1 + (0.859 + 0.859i)T + 13iT^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 + (-2.99 - 2.99i)T + 19iT^{2} \)
23 \( 1 + (-4.58 + 4.58i)T - 23iT^{2} \)
31 \( 1 + (6.55 - 6.55i)T - 31iT^{2} \)
37 \( 1 + 7.53iT - 37T^{2} \)
41 \( 1 + (6.89 + 6.89i)T + 41iT^{2} \)
43 \( 1 - 3.55iT - 43T^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 + (4.38 - 4.38i)T - 53iT^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 + (6.63 - 6.63i)T - 61iT^{2} \)
67 \( 1 + (7.77 - 7.77i)T - 67iT^{2} \)
71 \( 1 - 2.62iT - 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + (6.35 + 6.35i)T + 79iT^{2} \)
83 \( 1 + (9.67 - 9.67i)T - 83iT^{2} \)
89 \( 1 + (-3.97 - 3.97i)T + 89iT^{2} \)
97 \( 1 + 3.10iT - 97T^{2} \)
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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

−9.922214372148820861434672886378, −9.246932043626773457694269777919, −8.644446568197992178805242423345, −7.43396682181804223359138686400, −6.43802043683900453968648006845, −5.53293875806196257232064553226, −4.91008741768569943568495416725, −3.75462329930344451448800486786, −2.86671429234501950386830357915, −1.41015072464816808053354033900, 1.19622219461263720176179094662, 1.88244525214646897104388560076, 3.16792719834591629758952031794, 4.64564911296939164800955408669, 5.44070368436856608836990972547, 6.53857762455537899426649411699, 7.05759518323976449219273544833, 7.79000892698087761762220645494, 8.926763215232518134786070714226, 9.579704742387802245516146119072

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