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Properties

Label 2-760-8.5-c1-0-20
Degree $2$
Conductor $760$
Sign $-0.940 - 0.338i$
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  + (0.561 + 1.29i)2-s + 0.777i·3-s + (−1.36 + 1.45i)4-s + i·5-s + (−1.00 + 0.436i)6-s + 2.42·7-s + (−2.66 − 0.957i)8-s + 2.39·9-s + (−1.29 + 0.561i)10-s + 2.16i·11-s + (−1.13 − 1.06i)12-s + 3.60i·13-s + (1.36 + 3.14i)14-s − 0.777·15-s + (−0.252 − 3.99i)16-s − 3.85·17-s + ⋯
L(s)  = 1  + (0.397 + 0.917i)2-s + 0.448i·3-s + (−0.684 + 0.729i)4-s + 0.447i·5-s + (−0.412 + 0.178i)6-s + 0.917·7-s + (−0.940 − 0.338i)8-s + 0.798·9-s + (−0.410 + 0.177i)10-s + 0.652i·11-s + (−0.327 − 0.307i)12-s + 1.00i·13-s + (0.364 + 0.841i)14-s − 0.200·15-s + (−0.0630 − 0.998i)16-s − 0.934·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.940 - 0.338i$
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: \((2,\ 760,\ (\ :1/2),\ -0.940 - 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.315773 + 1.81018i\)
\(L(\frac12)\) \(\approx\) \(0.315773 + 1.81018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.561 - 1.29i)T \)
5 \( 1 - iT \)
19 \( 1 - iT \)
good3 \( 1 - 0.777iT - 3T^{2} \)
7 \( 1 - 2.42T + 7T^{2} \)
11 \( 1 - 2.16iT - 11T^{2} \)
13 \( 1 - 3.60iT - 13T^{2} \)
17 \( 1 + 3.85T + 17T^{2} \)
23 \( 1 - 5.10T + 23T^{2} \)
29 \( 1 - 0.0158iT - 29T^{2} \)
31 \( 1 + 9.21T + 31T^{2} \)
37 \( 1 + 11.4iT - 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 - 5.75iT - 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 2.84iT - 53T^{2} \)
59 \( 1 + 5.56iT - 59T^{2} \)
61 \( 1 - 1.31iT - 61T^{2} \)
67 \( 1 + 4.07iT - 67T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + 7.41T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 - 16.3iT - 83T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−10.86098075747023888051921280093, −9.496042751707357874487901794298, −9.128531374676142875073682300697, −7.87666648418221350555190078280, −7.18684939198374387767012298067, −6.49316263124565794040022214236, −5.18490553336892050578678713045, −4.49708290126206581651714769951, −3.72635802596941814708786391371, −2.02350849725625187675904849949, 0.883275893867414349699890236373, 1.95615745935863921240341975232, 3.27873445490173669224146810746, 4.51430390752739649427878771259, 5.16284825741872460851274390753, 6.23747553760841867531529783433, 7.46588420848021360913207153398, 8.413967414072176628631087226799, 9.127344996504584942652355420892, 10.16807780841901631376851320814

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