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Properties

Label 1-517-517.3-r0-0-0
Degree $1$
Conductor $517$
Sign $0.990 - 0.140i$
Analytic cond. $2.40093$
Root an. cond. $2.40093$
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  + (−0.702 − 0.711i)2-s + (0.824 + 0.565i)3-s + (−0.0136 + 0.999i)4-s + (−0.662 − 0.749i)5-s + (−0.176 − 0.984i)6-s + (−0.996 − 0.0818i)7-s + (0.721 − 0.692i)8-s + (0.360 + 0.932i)9-s + (−0.0682 + 0.997i)10-s + (−0.576 + 0.816i)12-s + (−0.868 − 0.496i)13-s + (0.641 + 0.767i)14-s + (−0.122 − 0.992i)15-s + (−0.999 − 0.0273i)16-s + (0.554 + 0.832i)17-s + (0.410 − 0.911i)18-s + ⋯
L(s)  = 1  + (−0.702 − 0.711i)2-s + (0.824 + 0.565i)3-s + (−0.0136 + 0.999i)4-s + (−0.662 − 0.749i)5-s + (−0.176 − 0.984i)6-s + (−0.996 − 0.0818i)7-s + (0.721 − 0.692i)8-s + (0.360 + 0.932i)9-s + (−0.0682 + 0.997i)10-s + (−0.576 + 0.816i)12-s + (−0.868 − 0.496i)13-s + (0.641 + 0.767i)14-s + (−0.122 − 0.992i)15-s + (−0.999 − 0.0273i)16-s + (0.554 + 0.832i)17-s + (0.410 − 0.911i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.990 - 0.140i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ 0.990 - 0.140i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9563955546 - 0.06774003403i\)
\(L(\frac12)\) \(\approx\) \(0.9563955546 - 0.06774003403i\)
\(L(1)\) \(\approx\) \(0.8206514657 - 0.1140510842i\)
\(L(1)\) \(\approx\) \(0.8206514657 - 0.1140510842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.702 - 0.711i)T \)
3 \( 1 + (0.824 + 0.565i)T \)
5 \( 1 + (-0.662 - 0.749i)T \)
7 \( 1 + (-0.996 - 0.0818i)T \)
13 \( 1 + (-0.868 - 0.496i)T \)
17 \( 1 + (0.554 + 0.832i)T \)
19 \( 1 + (0.976 - 0.216i)T \)
23 \( 1 + (0.460 + 0.887i)T \)
29 \( 1 + (0.410 - 0.911i)T \)
31 \( 1 + (0.792 + 0.609i)T \)
37 \( 1 + (0.641 - 0.767i)T \)
41 \( 1 + (0.881 - 0.472i)T \)
43 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (-0.435 + 0.900i)T \)
59 \( 1 + (0.946 - 0.321i)T \)
61 \( 1 + (0.927 - 0.373i)T \)
67 \( 1 + (0.854 + 0.519i)T \)
71 \( 1 + (-0.702 + 0.711i)T \)
73 \( 1 + (0.256 + 0.966i)T \)
79 \( 1 + (0.905 - 0.423i)T \)
83 \( 1 + (0.0409 + 0.999i)T \)
89 \( 1 + (-0.917 + 0.398i)T \)
97 \( 1 + (-0.999 + 0.0273i)T \)
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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

−23.752587146563304694058760925451, −22.92377035140472911420045029903, −22.23331119749540118636331257593, −20.67213497208905369032631809463, −19.75573106173042260320193733400, −19.25534476260794112135341286447, −18.57395744622455473956891783551, −17.96903313769936858560053532503, −16.56740944384443785865750585641, −15.97374901370370919077787294158, −14.946102049169473808308649832157, −14.42501747399760395145770816304, −13.554766448314332249169660076032, −12.34621019147653164292978914273, −11.43553764629898525957031306005, −9.98419416946429164558464348878, −9.58372948002832523271181067090, −8.450727411583159872793334033357, −7.54119505096583300026411233913, −6.946553672418919949992968875960, −6.24015091929845292476047461345, −4.685980821171437768085469002172, −3.23282236839960737318072936866, −2.470036973730307830843336255166, −0.797082343965243600724639810956, 0.96237694343117074063476080139, 2.504203071646978642048958706611, 3.401996403208336356246232027745, 4.11209579574091399111953069819, 5.31622407314631137365387215454, 7.194385770966964999228590608226, 7.8910228541733813052667254555, 8.753062878915236644401171881619, 9.64843797766936859154707080132, 10.085161118850093195574890268194, 11.29913154063302495373363943364, 12.37287849455414639823099446134, 12.9514929432324290147942115918, 13.91994311120622153048051905595, 15.3520360362849013692858318681, 15.89842007532358254545132870031, 16.7345174523185242298235650269, 17.51943047383689925574448767327, 19.04567423115297580765073700961, 19.41357751223469994484519469166, 20.01401062363295757012339381995, 20.77137476531459143688003761957, 21.63213952507565568907582964182, 22.39943865977981191780782827405, 23.4117839993027746103747065880

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