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Properties

Label 1-517-517.118-r1-0-0
Degree $1$
Conductor $517$
Sign $0.983 + 0.181i$
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  + (−0.0409 + 0.999i)2-s + (−0.894 − 0.447i)3-s + (−0.996 − 0.0818i)4-s + (0.360 − 0.932i)5-s + (0.484 − 0.874i)6-s + (−0.881 − 0.472i)7-s + (0.122 − 0.992i)8-s + (0.598 + 0.800i)9-s + (0.917 + 0.398i)10-s + (0.854 + 0.519i)12-s + (0.999 − 0.0273i)13-s + (0.507 − 0.861i)14-s + (−0.740 + 0.672i)15-s + (0.986 + 0.163i)16-s + (−0.927 + 0.373i)17-s + (−0.824 + 0.565i)18-s + ⋯
L(s)  = 1  + (−0.0409 + 0.999i)2-s + (−0.894 − 0.447i)3-s + (−0.996 − 0.0818i)4-s + (0.360 − 0.932i)5-s + (0.484 − 0.874i)6-s + (−0.881 − 0.472i)7-s + (0.122 − 0.992i)8-s + (0.598 + 0.800i)9-s + (0.917 + 0.398i)10-s + (0.854 + 0.519i)12-s + (0.999 − 0.0273i)13-s + (0.507 − 0.861i)14-s + (−0.740 + 0.672i)15-s + (0.986 + 0.163i)16-s + (−0.927 + 0.373i)17-s + (−0.824 + 0.565i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.983 + 0.181i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.983 + 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9690484821 + 0.08887382661i\)
\(L(\frac12)\) \(\approx\) \(0.9690484821 + 0.08887382661i\)
\(L(1)\) \(\approx\) \(0.6896078972 + 0.1127397195i\)
\(L(1)\) \(\approx\) \(0.6896078972 + 0.1127397195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.0409 + 0.999i)T \)
3 \( 1 + (-0.894 - 0.447i)T \)
5 \( 1 + (0.360 - 0.932i)T \)
7 \( 1 + (-0.881 - 0.472i)T \)
13 \( 1 + (0.999 - 0.0273i)T \)
17 \( 1 + (-0.927 + 0.373i)T \)
19 \( 1 + (-0.256 + 0.966i)T \)
23 \( 1 + (0.962 + 0.269i)T \)
29 \( 1 + (-0.824 + 0.565i)T \)
31 \( 1 + (-0.702 - 0.711i)T \)
37 \( 1 + (0.507 + 0.861i)T \)
41 \( 1 + (0.981 + 0.190i)T \)
43 \( 1 + (-0.854 + 0.519i)T \)
53 \( 1 + (0.905 - 0.423i)T \)
59 \( 1 + (-0.385 - 0.922i)T \)
61 \( 1 + (0.662 + 0.749i)T \)
67 \( 1 + (-0.990 - 0.136i)T \)
71 \( 1 + (0.0409 + 0.999i)T \)
73 \( 1 + (0.0136 - 0.999i)T \)
79 \( 1 + (0.868 + 0.496i)T \)
83 \( 1 + (0.969 - 0.243i)T \)
89 \( 1 + (-0.775 - 0.631i)T \)
97 \( 1 + (0.986 - 0.163i)T \)
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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

−22.85826388386233295693875671502, −22.446907335116671217234866856196, −21.67204841866717412337405702025, −21.10683111120452268736729386844, −19.94957202671836094367615610415, −18.97893070576716134919811525166, −18.26658829281704708517343072215, −17.7190618558230989770473264976, −16.66862103131295453977324104891, −15.61422677636742868500948205231, −14.832845084235280370097519217264, −13.46940790046071651363739085060, −12.95504642942942060070454826105, −11.8089271540188156992406987678, −10.96506954839990025444633026963, −10.604514470695826423077413403615, −9.39620072375927490445484522340, −8.98766848134300722906796106703, −7.09558612142283736903956688913, −6.20409092342345202538328766560, −5.32415238773842105959565695706, −4.085945493113932104596916489071, −3.19116211278823522780300751040, −2.17322648767855276323108160984, −0.58521068362658391053970988525, 0.54966189293518010156290278136, 1.55672042964164765358924973030, 3.750706199866863900811568579146, 4.67298196131483702662034455163, 5.76801778438616151476874621086, 6.26318708244595437099846124325, 7.21605926348464339700300506226, 8.263693888304231705573253853311, 9.20968498200100825904907260395, 10.1249463438490284552832527503, 11.1780275350564770437035728397, 12.59661396902713065612964493081, 13.125412177819837144887843546911, 13.58404810719179783918556059882, 15.0300257362703641637411512361, 16.13260448197459956982168930228, 16.54826712861373215292667161501, 17.18217724799798200804654971092, 18.07843155575536959740662081224, 18.84490965806688024556558054141, 19.82941024851980553260452819575, 21.01467725749992625467387394178, 22.05164316275435595146133545332, 22.78252601043714384983543070637, 23.531647557773409705344301887925

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