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Properties

Label 1-517-517.63-r1-0-0
Degree $1$
Conductor $517$
Sign $0.876 + 0.480i$
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  + (−0.986 + 0.163i)2-s + (−0.282 − 0.959i)3-s + (0.946 − 0.321i)4-s + (0.0954 − 0.995i)5-s + (0.435 + 0.900i)6-s + (0.385 + 0.922i)7-s + (−0.881 + 0.472i)8-s + (−0.839 + 0.542i)9-s + (0.0682 + 0.997i)10-s + (−0.576 − 0.816i)12-s + (−0.994 − 0.109i)13-s + (−0.531 − 0.847i)14-s + (−0.981 + 0.190i)15-s + (0.792 − 0.609i)16-s + (−0.0409 − 0.999i)17-s + (0.740 − 0.672i)18-s + ⋯
L(s)  = 1  + (−0.986 + 0.163i)2-s + (−0.282 − 0.959i)3-s + (0.946 − 0.321i)4-s + (0.0954 − 0.995i)5-s + (0.435 + 0.900i)6-s + (0.385 + 0.922i)7-s + (−0.881 + 0.472i)8-s + (−0.839 + 0.542i)9-s + (0.0682 + 0.997i)10-s + (−0.576 − 0.816i)12-s + (−0.994 − 0.109i)13-s + (−0.531 − 0.847i)14-s + (−0.981 + 0.190i)15-s + (0.792 − 0.609i)16-s + (−0.0409 − 0.999i)17-s + (0.740 − 0.672i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.876 + 0.480i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.876 + 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5678051039 + 0.1453896445i\)
\(L(\frac12)\) \(\approx\) \(0.5678051039 + 0.1453896445i\)
\(L(1)\) \(\approx\) \(0.5540479318 - 0.1509615982i\)
\(L(1)\) \(\approx\) \(0.5540479318 - 0.1509615982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.986 + 0.163i)T \)
3 \( 1 + (-0.282 - 0.959i)T \)
5 \( 1 + (0.0954 - 0.995i)T \)
7 \( 1 + (0.385 + 0.922i)T \)
13 \( 1 + (-0.994 - 0.109i)T \)
17 \( 1 + (-0.0409 - 0.999i)T \)
19 \( 1 + (-0.507 + 0.861i)T \)
23 \( 1 + (0.460 - 0.887i)T \)
29 \( 1 + (0.740 - 0.672i)T \)
31 \( 1 + (-0.999 + 0.0273i)T \)
37 \( 1 + (-0.531 + 0.847i)T \)
41 \( 1 + (-0.721 + 0.692i)T \)
43 \( 1 + (0.576 - 0.816i)T \)
53 \( 1 + (-0.176 + 0.984i)T \)
59 \( 1 + (-0.0136 + 0.999i)T \)
61 \( 1 + (0.969 - 0.243i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (0.986 + 0.163i)T \)
73 \( 1 + (-0.998 + 0.0546i)T \)
79 \( 1 + (0.484 + 0.874i)T \)
83 \( 1 + (-0.554 - 0.832i)T \)
89 \( 1 + (-0.917 - 0.398i)T \)
97 \( 1 + (0.792 + 0.609i)T \)
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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

−23.33641962397856270434687912147, −22.06309872154848472496172040485, −21.61810841100520774059057443785, −20.75156413089435149345528086115, −19.68576351245420740097021637442, −19.311070064498526748982850829173, −17.809955398833994905109199858683, −17.4736399484658200623815144464, −16.7232485010351712485665592205, −15.675203459497461027473146375125, −14.8926201562795743000437077300, −14.23149411840326771261434578758, −12.69941234692450044106165868391, −11.41211335926020188224279240908, −10.91727424922843098493482615983, −10.2430492930926879656215204391, −9.518605907544041989369784420028, −8.43229467547676737153376399679, −7.28315797189372098082395261656, −6.651008755871322933349744276969, −5.35591869973344341803485416516, −4.00941024637999140791891450135, −3.1336340066222580817612138751, −1.92020574472249502905239135726, −0.277254173504827445421103878666, 0.80446732033316708579394971159, 1.904148491676685185742349451382, 2.661170812697179169871881480610, 4.90997938310898412329332365358, 5.6324184754391333771092651293, 6.63912893802467863838024636185, 7.68618944245302582214658870803, 8.418310715800411726105068928832, 9.11034237461445554385877172595, 10.18457836353312267194465014905, 11.45484784338009524234270955309, 12.13671789955532415180607533514, 12.66447810173571879092331711729, 14.03467693875983085841142779306, 14.98585786793879915695355355866, 16.05080232517307693790075592328, 16.92114766888240581266145622227, 17.42518280639516062774968546759, 18.43550285806994346222258049608, 18.89925243302294784028287337511, 19.9173217872468469294809111300, 20.58741984177219190322865218912, 21.55132320662794011992573843577, 22.74968930575117916638417897671, 23.85988552012693896901798555691

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