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Properties

Label 2-760-8.5-c1-0-33
Degree $2$
Conductor $760$
Sign $0.383 - 0.923i$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  + (1.30 − 0.540i)2-s + 3.43i·3-s + (1.41 − 1.41i)4-s − i·5-s + (1.85 + 4.49i)6-s + 3.17·7-s + (1.08 − 2.61i)8-s − 8.80·9-s + (−0.540 − 1.30i)10-s + 3.55i·11-s + (4.85 + 4.86i)12-s + 3.89i·13-s + (4.14 − 1.71i)14-s + 3.43·15-s + (0.00555 − 3.99i)16-s + 2.19·17-s + ⋯
L(s)  = 1  + (0.924 − 0.382i)2-s + 1.98i·3-s + (0.707 − 0.706i)4-s − 0.447i·5-s + (0.758 + 1.83i)6-s + 1.19·7-s + (0.383 − 0.923i)8-s − 2.93·9-s + (−0.170 − 0.413i)10-s + 1.07i·11-s + (1.40 + 1.40i)12-s + 1.08i·13-s + (1.10 − 0.458i)14-s + 0.887·15-s + (0.00138 − 0.999i)16-s + 0.532·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.383 - 0.923i$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37282 + 1.58368i\)
\(L(\frac12)\) \(\approx\) \(2.37282 + 1.58368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.540i)T \)
5 \( 1 + iT \)
19 \( 1 + iT \)
good3 \( 1 - 3.43iT - 3T^{2} \)
7 \( 1 - 3.17T + 7T^{2} \)
11 \( 1 - 3.55iT - 11T^{2} \)
13 \( 1 - 3.89iT - 13T^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 - 4.27iT - 29T^{2} \)
31 \( 1 + 2.70T + 31T^{2} \)
37 \( 1 + 0.572iT - 37T^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 + 0.256iT - 43T^{2} \)
47 \( 1 + 7.32T + 47T^{2} \)
53 \( 1 + 7.59iT - 53T^{2} \)
59 \( 1 + 9.58iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 - 4.55T + 71T^{2} \)
73 \( 1 - 4.26T + 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−10.64216615273410020522669509140, −9.730947118659885987974939387289, −9.201096177205684173361963464233, −8.141964119581326958366887270670, −6.75264606414466507496243055809, −5.29092858838698489455294510947, −4.94766328944014676127774143739, −4.30575902021460724060809956996, −3.36397080053320765639190376014, −1.93249127535256640889307205024, 1.22406956312162043986550214901, 2.50665781935737595234223541906, 3.34887408256214427830909316346, 5.21307292249080843141819469692, 5.81725272173394570772506504503, 6.63894001858921040100689260649, 7.61587582628380255463825955259, 7.961145730531388198207536943260, 8.683178281881104563019853181665, 10.82792070770879979283471154079

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