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Properties

Label 1-517-517.24-r1-0-0
Degree $1$
Conductor $517$
Sign $0.751 + 0.659i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (0.937 + 0.347i)2-s + (0.986 − 0.163i)3-s + (0.758 + 0.652i)4-s + (0.998 + 0.0546i)5-s + (0.981 + 0.190i)6-s + (0.435 + 0.900i)7-s + (0.484 + 0.874i)8-s + (0.946 − 0.321i)9-s + (0.917 + 0.398i)10-s + (0.854 + 0.519i)12-s + (0.282 − 0.959i)13-s + (0.0954 + 0.995i)14-s + (0.994 − 0.109i)15-s + (0.149 + 0.988i)16-s + (−0.641 − 0.767i)17-s + (0.999 + 0.0273i)18-s + ⋯
L(s)  = 1  + (0.937 + 0.347i)2-s + (0.986 − 0.163i)3-s + (0.758 + 0.652i)4-s + (0.998 + 0.0546i)5-s + (0.981 + 0.190i)6-s + (0.435 + 0.900i)7-s + (0.484 + 0.874i)8-s + (0.946 − 0.321i)9-s + (0.917 + 0.398i)10-s + (0.854 + 0.519i)12-s + (0.282 − 0.959i)13-s + (0.0954 + 0.995i)14-s + (0.994 − 0.109i)15-s + (0.149 + 0.988i)16-s + (−0.641 − 0.767i)17-s + (0.999 + 0.0273i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.997552997 + 2.632750664i\)
\(L(\frac12)\) \(\approx\) \(6.997552997 + 2.632750664i\)
\(L(1)\) \(\approx\) \(3.199030800 + 0.8094284514i\)
\(L(1)\) \(\approx\) \(3.199030800 + 0.8094284514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.937 + 0.347i)T \)
3 \( 1 + (0.986 - 0.163i)T \)
5 \( 1 + (0.998 + 0.0546i)T \)
7 \( 1 + (0.435 + 0.900i)T \)
13 \( 1 + (0.282 - 0.959i)T \)
17 \( 1 + (-0.641 - 0.767i)T \)
19 \( 1 + (-0.360 - 0.932i)T \)
23 \( 1 + (0.962 + 0.269i)T \)
29 \( 1 + (0.999 + 0.0273i)T \)
31 \( 1 + (-0.894 + 0.447i)T \)
37 \( 1 + (0.0954 - 0.995i)T \)
41 \( 1 + (-0.905 + 0.423i)T \)
43 \( 1 + (-0.854 + 0.519i)T \)
53 \( 1 + (-0.122 - 0.992i)T \)
59 \( 1 + (-0.230 + 0.973i)T \)
61 \( 1 + (-0.507 + 0.861i)T \)
67 \( 1 + (-0.990 - 0.136i)T \)
71 \( 1 + (-0.937 + 0.347i)T \)
73 \( 1 + (-0.598 + 0.800i)T \)
79 \( 1 + (0.740 - 0.672i)T \)
83 \( 1 + (0.531 + 0.847i)T \)
89 \( 1 + (-0.775 - 0.631i)T \)
97 \( 1 + (0.149 - 0.988i)T \)
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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

−23.39219107471032909489275463916, −22.05839984906503893178371145858, −21.51059342237478398865586986511, −20.709246032030664008481864648902, −20.31815269950519372349374393103, −19.23534345245641067142980289965, −18.49204653552358543814332966216, −17.09104374594483199641673978675, −16.38109041703655534614901554685, −15.09662876984590629580149104537, −14.50928097532184681644956839500, −13.665975386884815516034108831718, −13.34389164182598148131579909000, −12.26895441171596966410345617700, −10.86314562016274729267245872841, −10.34469574235098649062335086134, −9.36927327351482650799317362553, −8.31744239604216746335164856350, −7.02036595640902159237038196954, −6.31337283130972972327619546284, −4.90777500888055393410069383478, −4.18811259289844551198546915622, −3.20703653877906341145316639644, −1.94741924986809680979536561634, −1.430478207378214240274288178523, 1.52640497321959825880111792977, 2.58657942358660374652792839222, 3.076788738616936755156507043111, 4.648398128282869954838625618019, 5.39180156703594455668336239394, 6.483851619253385822359403372889, 7.33072803395176359869205169836, 8.53730484066377614243944292964, 9.0809418135772440650909774095, 10.41393073426473541584301227351, 11.47097925533479003504734973722, 12.69950790863972711238886892183, 13.217237085745933256920637042448, 13.970295942482867292680398727, 14.89813751600221261258411901227, 15.336064336574527959304218491460, 16.348020973063161329933649662051, 17.774267610924993540678061914787, 18.06338458344410941985581871499, 19.4761088846902630738268995221, 20.33320585476857275307591503039, 21.1385494493726973541769681927, 21.63693278402347773783402560203, 22.428675571901460958556427149869, 23.57877127639895813000597759109

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