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Properties

Label 1-517-517.84-r1-0-0
Degree $1$
Conductor $517$
Sign $0.600 - 0.799i$
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  + (0.662 + 0.749i)2-s + (0.641 − 0.767i)3-s + (−0.122 + 0.992i)4-s + (−0.230 − 0.973i)5-s + (0.999 − 0.0273i)6-s + (0.740 + 0.672i)7-s + (−0.824 + 0.565i)8-s + (−0.176 − 0.984i)9-s + (0.576 − 0.816i)10-s + (0.682 + 0.730i)12-s + (−0.0409 − 0.999i)13-s + (−0.0136 + 0.999i)14-s + (−0.894 − 0.447i)15-s + (−0.969 − 0.243i)16-s + (0.839 − 0.542i)17-s + (0.620 − 0.784i)18-s + ⋯
L(s)  = 1  + (0.662 + 0.749i)2-s + (0.641 − 0.767i)3-s + (−0.122 + 0.992i)4-s + (−0.230 − 0.973i)5-s + (0.999 − 0.0273i)6-s + (0.740 + 0.672i)7-s + (−0.824 + 0.565i)8-s + (−0.176 − 0.984i)9-s + (0.576 − 0.816i)10-s + (0.682 + 0.730i)12-s + (−0.0409 − 0.999i)13-s + (−0.0136 + 0.999i)14-s + (−0.894 − 0.447i)15-s + (−0.969 − 0.243i)16-s + (0.839 − 0.542i)17-s + (0.620 − 0.784i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.600 - 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.039442722 - 1.517830142i\)
\(L(\frac12)\) \(\approx\) \(3.039442722 - 1.517830142i\)
\(L(1)\) \(\approx\) \(1.819361343 - 0.08844745195i\)
\(L(1)\) \(\approx\) \(1.819361343 - 0.08844745195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.662 + 0.749i)T \)
3 \( 1 + (0.641 - 0.767i)T \)
5 \( 1 + (-0.230 - 0.973i)T \)
7 \( 1 + (0.740 + 0.672i)T \)
13 \( 1 + (-0.0409 - 0.999i)T \)
17 \( 1 + (0.839 - 0.542i)T \)
19 \( 1 + (0.385 + 0.922i)T \)
23 \( 1 + (-0.917 - 0.398i)T \)
29 \( 1 + (0.620 - 0.784i)T \)
31 \( 1 + (0.927 - 0.373i)T \)
37 \( 1 + (-0.0136 - 0.999i)T \)
41 \( 1 + (0.282 - 0.959i)T \)
43 \( 1 + (-0.682 + 0.730i)T \)
53 \( 1 + (0.792 - 0.609i)T \)
59 \( 1 + (-0.981 - 0.190i)T \)
61 \( 1 + (0.955 - 0.295i)T \)
67 \( 1 + (0.203 - 0.979i)T \)
71 \( 1 + (-0.662 + 0.749i)T \)
73 \( 1 + (-0.721 + 0.692i)T \)
79 \( 1 + (0.702 - 0.711i)T \)
83 \( 1 + (-0.360 - 0.932i)T \)
89 \( 1 + (0.854 - 0.519i)T \)
97 \( 1 + (-0.969 + 0.243i)T \)
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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

−23.46643119794396889419652332622, −22.336592359290134642378310397701, −21.65362604861366370789645569764, −21.15344687996909506072730820930, −20.11212375993793476732662482801, −19.55693394377028231952286771873, −18.72668324678310723794844133969, −17.733462347810127747336119268621, −16.4402426013055420190426251269, −15.40303118985090356307242768461, −14.71404180404180107248119973827, −13.99637210623815620272997133558, −13.58629241631699144506866378091, −11.93489278977413836611840472942, −11.30032264240078301898027850585, −10.38862436749499076285163861494, −9.91371014395647161835564700439, −8.6713723174007191496757650710, −7.54296424011677139053133808430, −6.46160425246396301287685926248, −5.07988350359580425707925405510, −4.25049907848179169980258455796, −3.47247265798063305885526726666, −2.54970823190694597036976346865, −1.40226781132783009429222965762, 0.62426275343191003226396315669, 2.04328539637248811347381008724, 3.179676584538359330963652061864, 4.30124455957527156649519444567, 5.42116557721107402922957009336, 6.045421051822208671751159766253, 7.55653494160238872456370952061, 8.047628995176564417307878255178, 8.639441486729054912385713172486, 9.77759142983466167640713084397, 11.75902541213199626279942951364, 12.16555994888931504088204012135, 12.91352047185922645195626144817, 13.881617894586400906430570488805, 14.54450995509267112190295958770, 15.45495503613645973662353644133, 16.1882158217072534682714735722, 17.346746611102486153505232849559, 17.967778416802295759788388229847, 18.88659607643588093955447005832, 20.13512748672840151792220842763, 20.76249782766373738307306393032, 21.36646892797368307064679522160, 22.73462035504689674340791557361, 23.3566192434582924701834907426

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