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Properties

Label 1-517-517.241-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.999 + 0.0138i$
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  + (−0.682 + 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (0.460 − 0.887i)5-s + (0.775 − 0.631i)6-s + (0.917 − 0.398i)7-s + (0.775 + 0.631i)8-s + (0.962 + 0.269i)9-s + (0.334 + 0.942i)10-s + (−0.0682 + 0.997i)12-s + (−0.854 − 0.519i)13-s + (−0.334 + 0.942i)14-s + (−0.576 + 0.816i)15-s + (−0.990 + 0.136i)16-s + (−0.203 − 0.979i)17-s + (−0.854 + 0.519i)18-s + ⋯
L(s)  = 1  + (−0.682 + 0.730i)2-s + (−0.990 − 0.136i)3-s + (−0.0682 − 0.997i)4-s + (0.460 − 0.887i)5-s + (0.775 − 0.631i)6-s + (0.917 − 0.398i)7-s + (0.775 + 0.631i)8-s + (0.962 + 0.269i)9-s + (0.334 + 0.942i)10-s + (−0.0682 + 0.997i)12-s + (−0.854 − 0.519i)13-s + (−0.334 + 0.942i)14-s + (−0.576 + 0.816i)15-s + (−0.990 + 0.136i)16-s + (−0.203 − 0.979i)17-s + (−0.854 + 0.519i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.999 + 0.0138i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.999 + 0.0138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002214673069 - 0.3205048302i\)
\(L(\frac12)\) \(\approx\) \(0.002214673069 - 0.3205048302i\)
\(L(1)\) \(\approx\) \(0.5631113696 - 0.07277155159i\)
\(L(1)\) \(\approx\) \(0.5631113696 - 0.07277155159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.682 + 0.730i)T \)
3 \( 1 + (-0.990 - 0.136i)T \)
5 \( 1 + (0.460 - 0.887i)T \)
7 \( 1 + (0.917 - 0.398i)T \)
13 \( 1 + (-0.854 - 0.519i)T \)
17 \( 1 + (-0.203 - 0.979i)T \)
19 \( 1 + (-0.460 - 0.887i)T \)
23 \( 1 + (0.682 + 0.730i)T \)
29 \( 1 + (-0.854 + 0.519i)T \)
31 \( 1 + (-0.990 + 0.136i)T \)
37 \( 1 + (-0.334 - 0.942i)T \)
41 \( 1 + (0.775 - 0.631i)T \)
43 \( 1 + (0.0682 + 0.997i)T \)
53 \( 1 + (-0.775 + 0.631i)T \)
59 \( 1 + (-0.0682 + 0.997i)T \)
61 \( 1 + (0.334 - 0.942i)T \)
67 \( 1 + (-0.917 - 0.398i)T \)
71 \( 1 + (0.682 + 0.730i)T \)
73 \( 1 + (-0.962 + 0.269i)T \)
79 \( 1 + (0.576 - 0.816i)T \)
83 \( 1 + (-0.203 + 0.979i)T \)
89 \( 1 + (0.460 - 0.887i)T \)
97 \( 1 + (-0.990 - 0.136i)T \)
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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

−23.693770010603809747939172552425, −22.50063504791132827336114639852, −22.02357478955704515616956697514, −21.29139113802453423910630391159, −20.671088303101742420723212902704, −19.13052642906694293428215918520, −18.70666713198982934151421583115, −17.85455788958304384743002988878, −17.19000330965240142114300179797, −16.61478435942327664094847890300, −15.19083627213759414910679147222, −14.4844416019743422380446454142, −13.09554114742479333706058602497, −12.24453241036705498484612898470, −11.40555015439133652500545602069, −10.76846682797465675369640633329, −10.07918285380620145761237750366, −9.11745217507152481452304436075, −7.89308078346822906339690129589, −6.970452143327995491331736220616, −5.977048892584641183982263531207, −4.77900174018706769455411852362, −3.750554406989273370906766236382, −2.25671777338148582280129171269, −1.564526326837429679415659935234, 0.13257329793386959040203226923, 1.03862032957814251982585757958, 2.0490817381307505283650430512, 4.52106877998949192008058719298, 5.108252609342070279708249614948, 5.768163859230590133407945479300, 7.1383858694701143384473867493, 7.59074834568895802369581691905, 8.92119545855747154498859708956, 9.62405056569960836965599817215, 10.745831029548749141646764969654, 11.34552041797965940809861959668, 12.59476174162852448242336575557, 13.459433794865934269921501506368, 14.489771398243184835401229107475, 15.51933608199010673527634971707, 16.384156323230619719176748383963, 17.13381852616689135501394330608, 17.62622259168873720690516321264, 18.20677715828828916027956965577, 19.45434623051610156219060882019, 20.29329886044684874484999074782, 21.24995207938164590944212155302, 22.23288774823963372504234330537, 23.25279906590608285528917680017

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