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Properties

Label 1-517-517.72-r1-0-0
Degree $1$
Conductor $517$
Sign $0.290 + 0.956i$
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  + (−0.410 − 0.911i)2-s + (0.905 + 0.423i)3-s + (−0.662 + 0.749i)4-s + (−0.620 − 0.784i)5-s + (0.0136 − 0.999i)6-s + (−0.360 − 0.932i)7-s + (0.955 + 0.295i)8-s + (0.641 + 0.767i)9-s + (−0.460 + 0.887i)10-s + (−0.917 + 0.398i)12-s + (−0.721 + 0.692i)13-s + (−0.702 + 0.711i)14-s + (−0.230 − 0.973i)15-s + (−0.122 − 0.992i)16-s + (0.282 − 0.959i)17-s + (0.435 − 0.900i)18-s + ⋯
L(s)  = 1  + (−0.410 − 0.911i)2-s + (0.905 + 0.423i)3-s + (−0.662 + 0.749i)4-s + (−0.620 − 0.784i)5-s + (0.0136 − 0.999i)6-s + (−0.360 − 0.932i)7-s + (0.955 + 0.295i)8-s + (0.641 + 0.767i)9-s + (−0.460 + 0.887i)10-s + (−0.917 + 0.398i)12-s + (−0.721 + 0.692i)13-s + (−0.702 + 0.711i)14-s + (−0.230 − 0.973i)15-s + (−0.122 − 0.992i)16-s + (0.282 − 0.959i)17-s + (0.435 − 0.900i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.290 + 0.956i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.290 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3395490634 + 0.2518357207i\)
\(L(\frac12)\) \(\approx\) \(0.3395490634 + 0.2518357207i\)
\(L(1)\) \(\approx\) \(0.7390993315 - 0.3024350495i\)
\(L(1)\) \(\approx\) \(0.7390993315 - 0.3024350495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.410 - 0.911i)T \)
3 \( 1 + (0.905 + 0.423i)T \)
5 \( 1 + (-0.620 - 0.784i)T \)
7 \( 1 + (-0.360 - 0.932i)T \)
13 \( 1 + (-0.721 + 0.692i)T \)
17 \( 1 + (0.282 - 0.959i)T \)
19 \( 1 + (-0.554 - 0.832i)T \)
23 \( 1 + (0.203 + 0.979i)T \)
29 \( 1 + (0.435 - 0.900i)T \)
31 \( 1 + (-0.981 - 0.190i)T \)
37 \( 1 + (-0.702 - 0.711i)T \)
41 \( 1 + (-0.598 + 0.800i)T \)
43 \( 1 + (0.917 + 0.398i)T \)
53 \( 1 + (0.946 + 0.321i)T \)
59 \( 1 + (0.0954 + 0.995i)T \)
61 \( 1 + (-0.149 + 0.988i)T \)
67 \( 1 + (-0.775 - 0.631i)T \)
71 \( 1 + (0.410 - 0.911i)T \)
73 \( 1 + (-0.927 - 0.373i)T \)
79 \( 1 + (0.385 - 0.922i)T \)
83 \( 1 + (-0.824 + 0.565i)T \)
89 \( 1 + (0.962 + 0.269i)T \)
97 \( 1 + (-0.122 + 0.992i)T \)
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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

−23.40921492211350096926396647999, −22.47016017689447591448467028401, −21.739646399923987656358098081726, −20.3194483022578367893017968750, −19.41381705465706889565790602134, −18.8788639757616643414990052006, −18.341671801114131033632600881238, −17.33715159059275832621655367154, −16.12725391360432789223682768992, −15.286036876511729379385041804190, −14.761350695687423532526376220167, −14.211100856002525504313417067167, −12.798540237921825761937975113691, −12.290299385984336811258966937938, −10.60384351166822137697137028014, −9.92130289327127520185870583019, −8.66974686749973635324931981049, −8.25750307589628783568132347060, −7.23006017804682963274696582124, −6.53516898800704218560324571945, −5.49215924185383942304030473213, −4.0265054312593597492681764888, −3.00228102232033008506722877134, −1.834106088519183188350665616487, −0.11923301576912541384424075142, 1.094789122636082627180526592075, 2.376608826700094948981061937563, 3.47166902333052235408561180496, 4.265583513809285998849514967583, 4.93861458565799150816035348998, 7.26991983770744937555430506366, 7.68397736201372053234227743280, 8.96666197609348282297370978940, 9.36361512854211933769060807706, 10.30990833493498419176246542704, 11.311133796493204642917497952791, 12.21854318969076571058787130901, 13.29191379852304031541631302434, 13.71267607317846875701264712003, 14.90966332223173277670735012777, 16.111697267500254131274795667634, 16.64972414408357235647268358110, 17.57852190700552052900003342461, 18.93541094861688084129355702804, 19.637147915849341029922713225520, 19.916619820113676737054208581725, 20.900928009146023520342992978487, 21.40061741672569935697701139555, 22.51235129183565404927617012263, 23.42024667531785240194649409121

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