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Properties

Label 1-517-517.105-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.789 - 0.613i$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
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  + (−0.641 − 0.767i)2-s + (0.0409 + 0.999i)3-s + (−0.176 + 0.984i)4-s + (0.0136 − 0.999i)5-s + (0.740 − 0.672i)6-s + (0.484 + 0.874i)7-s + (0.868 − 0.496i)8-s + (−0.996 + 0.0818i)9-s + (−0.775 + 0.631i)10-s + (−0.990 − 0.136i)12-s + (−0.894 − 0.447i)13-s + (0.360 − 0.932i)14-s + (0.999 − 0.0273i)15-s + (−0.937 − 0.347i)16-s + (−0.976 − 0.216i)17-s + (0.702 + 0.711i)18-s + ⋯
L(s)  = 1  + (−0.641 − 0.767i)2-s + (0.0409 + 0.999i)3-s + (−0.176 + 0.984i)4-s + (0.0136 − 0.999i)5-s + (0.740 − 0.672i)6-s + (0.484 + 0.874i)7-s + (0.868 − 0.496i)8-s + (−0.996 + 0.0818i)9-s + (−0.775 + 0.631i)10-s + (−0.990 − 0.136i)12-s + (−0.894 − 0.447i)13-s + (0.360 − 0.932i)14-s + (0.999 − 0.0273i)15-s + (−0.937 − 0.347i)16-s + (−0.976 − 0.216i)17-s + (0.702 + 0.711i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.789 - 0.613i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1111650517 - 0.3242382164i\)
\(L(\frac12)\) \(\approx\) \(0.1111650517 - 0.3242382164i\)
\(L(1)\) \(\approx\) \(0.5835236295 - 0.1224358034i\)
\(L(1)\) \(\approx\) \(0.5835236295 - 0.1224358034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.641 - 0.767i)T \)
3 \( 1 + (0.0409 + 0.999i)T \)
5 \( 1 + (0.0136 - 0.999i)T \)
7 \( 1 + (0.484 + 0.874i)T \)
13 \( 1 + (-0.894 - 0.447i)T \)
17 \( 1 + (-0.976 - 0.216i)T \)
19 \( 1 + (-0.955 - 0.295i)T \)
23 \( 1 + (0.0682 - 0.997i)T \)
29 \( 1 + (-0.702 - 0.711i)T \)
31 \( 1 + (0.620 - 0.784i)T \)
37 \( 1 + (0.360 + 0.932i)T \)
41 \( 1 + (0.994 - 0.109i)T \)
43 \( 1 + (-0.990 + 0.136i)T \)
53 \( 1 + (0.410 + 0.911i)T \)
59 \( 1 + (-0.435 + 0.900i)T \)
61 \( 1 + (-0.256 - 0.966i)T \)
67 \( 1 + (-0.682 - 0.730i)T \)
71 \( 1 + (0.641 - 0.767i)T \)
73 \( 1 + (-0.230 - 0.973i)T \)
79 \( 1 + (-0.824 - 0.565i)T \)
83 \( 1 + (-0.507 - 0.861i)T \)
89 \( 1 + (-0.576 - 0.816i)T \)
97 \( 1 + (-0.937 + 0.347i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.92507421919626256520312290807, −23.34810917815119813352538374886, −22.616318526099532653898525762966, −21.45305404261535494822271941074, −19.93184710067510033546333890743, −19.52830229047599810737828785648, −18.71895126585553912971591204509, −17.750012722888760450359691687766, −17.44864393374721160721403590988, −16.49932389100981461928845035900, −15.15448289650888194280290460933, −14.47989512587207733168617459253, −13.89387635868787742532988072947, −12.93783723429901794710206796748, −11.457595241548965751706096928088, −10.871952771936297406993521348836, −9.87743755482391580923068988437, −8.69824802493018984851148063466, −7.74043150342174631579903349488, −7.05826715786151080876721974407, −6.561334607259611358763093799260, −5.39704425596280278075459050378, −4.05843887165416878775057112676, −2.40355236277238900684096721565, −1.496225182603810169033498124880, 0.21684299714508798708780763914, 2.04270513933040786845670849961, 2.82316717057743657227761305959, 4.433301999614711047142617493260, 4.68813007608183974425178792329, 6.04180018673893460100706238128, 7.83139108626209939379768876744, 8.606355903265474119639974866598, 9.18873387713727618492515851484, 9.99887332461078554134604289022, 11.02415554834769169666613046809, 11.80067117607779132294906844744, 12.59528571114203715178443124469, 13.551197524633699146424457773761, 14.97195371891312550902969982499, 15.577449231725225946105578873071, 16.73244564886901552907200199726, 17.160336310101293909057018721, 18.07755746795329673866847189895, 19.22295062946427372638547966333, 20.05292028451557470670967322010, 20.66424240016147868998937218660, 21.39314296910475387702457100311, 22.02650043327442547476619370412, 22.8240676452336390129844902654

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