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Properties

Label 2-760-8.5-c1-0-13
Degree $2$
Conductor $760$
Sign $0.684 - 0.729i$
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  + (−1.01 − 0.989i)2-s − 0.530i·3-s + (0.0426 + 1.99i)4-s + i·5-s + (−0.525 + 0.536i)6-s + 1.59·7-s + (1.93 − 2.06i)8-s + 2.71·9-s + (0.989 − 1.01i)10-s + 4.08i·11-s + (1.06 − 0.0226i)12-s + 4.29i·13-s + (−1.61 − 1.58i)14-s + 0.530·15-s + (−3.99 + 0.170i)16-s − 7.51·17-s + ⋯
L(s)  = 1  + (−0.714 − 0.699i)2-s − 0.306i·3-s + (0.0213 + 0.999i)4-s + 0.447i·5-s + (−0.214 + 0.219i)6-s + 0.604·7-s + (0.684 − 0.729i)8-s + 0.906·9-s + (0.312 − 0.319i)10-s + 1.23i·11-s + (0.306 − 0.00653i)12-s + 1.19i·13-s + (−0.431 − 0.422i)14-s + 0.137·15-s + (−0.999 + 0.0426i)16-s − 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.684 - 0.729i$
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: \((2,\ 760,\ (\ :1/2),\ 0.684 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874947 + 0.378920i\)
\(L(\frac12)\) \(\approx\) \(0.874947 + 0.378920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.01 + 0.989i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 + 0.530iT - 3T^{2} \)
7 \( 1 - 1.59T + 7T^{2} \)
11 \( 1 - 4.08iT - 11T^{2} \)
13 \( 1 - 4.29iT - 13T^{2} \)
17 \( 1 + 7.51T + 17T^{2} \)
23 \( 1 + 7.97T + 23T^{2} \)
29 \( 1 - 5.77iT - 29T^{2} \)
31 \( 1 - 3.57T + 31T^{2} \)
37 \( 1 - 1.10iT - 37T^{2} \)
41 \( 1 - 2.90T + 41T^{2} \)
43 \( 1 + 4.31iT - 43T^{2} \)
47 \( 1 - 6.15T + 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 6.91iT - 59T^{2} \)
61 \( 1 - 4.61iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 + 3.08T + 79T^{2} \)
83 \( 1 + 3.40iT - 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 3.07T + 97T^{2} \)
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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

−10.44640312264591195284298509635, −9.660688501652673653112817598734, −8.905793457628784836962662331815, −7.87237134750806433152591312455, −7.10293138772722815859678848985, −6.52327204915914744671866761280, −4.53235808818573427222908284736, −4.10270858684301189278457392454, −2.27316782650361861038652884422, −1.70866474077615257360051665354, 0.61726254593048996314239163059, 2.15113858254125142445458750921, 4.04283028546845120022314502625, 4.94161194756588215092936606672, 5.89483462982306575942020506261, 6.73710980876080815483893596396, 8.107932916222089077467792512469, 8.187535593259452632021157694615, 9.329376908290471219250103560635, 10.06015202576458028824243255856

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