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Properties

Label 2-1160-145.17-c1-0-34
Degree $2$
Conductor $1160$
Sign $0.538 + 0.842i$
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  + 3.17i·3-s + (1.08 − 1.95i)5-s + (−1.73 − 1.73i)7-s − 7.05·9-s + (0.415 − 0.415i)11-s + (−2.82 − 2.82i)13-s + (6.20 + 3.43i)15-s − 0.788·17-s + (−0.544 − 0.544i)19-s + (5.50 − 5.50i)21-s + (3.37 − 3.37i)23-s + (−2.64 − 4.24i)25-s − 12.8i·27-s + (−4.07 + 3.51i)29-s + (4.85 − 4.85i)31-s + ⋯
L(s)  = 1  + 1.83i·3-s + (0.484 − 0.874i)5-s + (−0.655 − 0.655i)7-s − 2.35·9-s + (0.125 − 0.125i)11-s + (−0.782 − 0.782i)13-s + (1.60 + 0.887i)15-s − 0.191·17-s + (−0.124 − 0.124i)19-s + (1.20 − 1.20i)21-s + (0.702 − 0.702i)23-s + (−0.529 − 0.848i)25-s − 2.47i·27-s + (−0.757 + 0.652i)29-s + (0.871 − 0.871i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $0.538 + 0.842i$
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: \((2,\ 1160,\ (\ :1/2),\ 0.538 + 0.842i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9082754031\)
\(L(\frac12)\) \(\approx\) \(0.9082754031\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.08 + 1.95i)T \)
29 \( 1 + (4.07 - 3.51i)T \)
good3 \( 1 - 3.17iT - 3T^{2} \)
7 \( 1 + (1.73 + 1.73i)T + 7iT^{2} \)
11 \( 1 + (-0.415 + 0.415i)T - 11iT^{2} \)
13 \( 1 + (2.82 + 2.82i)T + 13iT^{2} \)
17 \( 1 + 0.788T + 17T^{2} \)
19 \( 1 + (0.544 + 0.544i)T + 19iT^{2} \)
23 \( 1 + (-3.37 + 3.37i)T - 23iT^{2} \)
31 \( 1 + (-4.85 + 4.85i)T - 31iT^{2} \)
37 \( 1 + 5.85iT - 37T^{2} \)
41 \( 1 + (4.77 + 4.77i)T + 41iT^{2} \)
43 \( 1 - 1.81iT - 43T^{2} \)
47 \( 1 + 8.00iT - 47T^{2} \)
53 \( 1 + (-2.27 + 2.27i)T - 53iT^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 + (6.91 - 6.91i)T - 61iT^{2} \)
67 \( 1 + (-6.03 + 6.03i)T - 67iT^{2} \)
71 \( 1 - 8.30iT - 71T^{2} \)
73 \( 1 + 7.30T + 73T^{2} \)
79 \( 1 + (-8.17 - 8.17i)T + 79iT^{2} \)
83 \( 1 + (0.959 - 0.959i)T - 83iT^{2} \)
89 \( 1 + (6.07 + 6.07i)T + 89iT^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−9.825253543603715286011930167668, −9.002724622056279354564496571261, −8.469446179864810782822656898860, −7.17333633122386095052872395768, −5.91235276218657768508131961229, −5.21936644987594153669047613746, −4.45679133677247759835704743623, −3.67893437890769910679844741132, −2.60644550152228547928098500445, −0.37562779832790565567475209398, 1.57842573445315071434989061195, 2.46259160896529282829668442927, 3.21826343565598531171705695463, 5.09416339248495005092020750349, 6.23290030420269312670398823315, 6.51421750735580754359233102700, 7.28208733668507235023701827772, 8.024880607095235067283143322310, 9.116295805957906518704650815183, 9.717859883302993687946356186872

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