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Properties

Label 2-760-760.539-c0-0-1
Degree $2$
Conductor $760$
Sign $-0.305 - 0.952i$
Analytic cond. $0.379289$
Root an. cond. $0.615864$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)10-s − 11-s + (1 − 1.73i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 0.999·18-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 7-s − 0.999·8-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)10-s − 11-s + (1 − 1.73i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 0.999·18-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(0.379289\)
Root analytic conductor: \(0.615864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 760,\ (\ :0),\ -0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267871686\)
\(L(\frac12)\) \(\approx\) \(1.267871686\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.93485441171755481527707135520, −10.06825674519893421042825104863, −8.727327446560190403340937227162, −7.88324086271870167173093584438, −7.59851929498179832566232844072, −6.18683273596882901435283575561, −5.53957220539833212944099355730, −4.82679929257341686661846767837, −3.33634997677515689541557517985, −2.40768412127471461183800674231, 1.37604636577870496423637973947, 2.40058467866563171112762691553, 4.01744006617182300204703109441, 4.60651403833724601930028635338, 5.74952792046197916440694269206, 6.31742644072130962741723524899, 8.058704218817334551105236586684, 8.830312691135980401438075322715, 9.394745800195968114513699742530, 10.48250121262372326865424083650

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