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Properties

Label 2-760-8.5-c1-0-46
Degree $2$
Conductor $760$
Sign $0.988 + 0.152i$
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  + (1.41 + 0.0719i)2-s − 1.16i·3-s + (1.98 + 0.203i)4-s + i·5-s + (0.0836 − 1.64i)6-s + 0.494·7-s + (2.79 + 0.430i)8-s + 1.64·9-s + (−0.0719 + 1.41i)10-s + 3.06i·11-s + (0.236 − 2.31i)12-s − 1.31i·13-s + (0.698 + 0.0355i)14-s + 1.16·15-s + (3.91 + 0.808i)16-s + 0.965·17-s + ⋯
L(s)  = 1  + (0.998 + 0.0508i)2-s − 0.671i·3-s + (0.994 + 0.101i)4-s + 0.447i·5-s + (0.0341 − 0.670i)6-s + 0.186·7-s + (0.988 + 0.152i)8-s + 0.549·9-s + (−0.0227 + 0.446i)10-s + 0.924i·11-s + (0.0682 − 0.667i)12-s − 0.365i·13-s + (0.186 + 0.00950i)14-s + 0.300·15-s + (0.979 + 0.202i)16-s + 0.234·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.988 + 0.152i$
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: \((2,\ 760,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11138 - 0.237971i\)
\(L(\frac12)\) \(\approx\) \(3.11138 - 0.237971i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0719i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 + 1.16iT - 3T^{2} \)
7 \( 1 - 0.494T + 7T^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
13 \( 1 + 1.31iT - 13T^{2} \)
17 \( 1 - 0.965T + 17T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 + 6.53iT - 29T^{2} \)
31 \( 1 - 8.53T + 31T^{2} \)
37 \( 1 + 0.835iT - 37T^{2} \)
41 \( 1 - 3.67T + 41T^{2} \)
43 \( 1 - 7.66iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + 11.7iT - 53T^{2} \)
59 \( 1 - 1.11iT - 59T^{2} \)
61 \( 1 + 3.52iT - 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 + 7.03T + 71T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 16.3iT - 83T^{2} \)
89 \( 1 + 0.454T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−10.18444709352778238001134486673, −9.958283160710969538454532295070, −8.011359880977951657863821703085, −7.68513098353523911541581160281, −6.61864010753992019968702749404, −6.08467336081711118361460510442, −4.78571651317468477610246572229, −3.97302197211749677388588675417, −2.62562032883470384142813967677, −1.63543632483110899163644327843, 1.52587461733381853736912208547, 3.07311747155891620286824690397, 4.08287496556439766173043169522, 4.76884412786879417583099327799, 5.71336648075314117085799983943, 6.60874501699885659394789846867, 7.70853337600671601011762988699, 8.628931016547224906613185441905, 9.743724234465492138172637547995, 10.45332955445195345800543307984

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