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Properties

Label 1-517-517.2-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.880 - 0.473i$
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  + (0.620 + 0.784i)2-s + (−0.702 − 0.711i)3-s + (−0.230 + 0.973i)4-s + (0.256 − 0.966i)5-s + (0.122 − 0.992i)6-s + (0.176 + 0.984i)7-s + (−0.905 + 0.423i)8-s + (−0.0136 + 0.999i)9-s + (0.917 − 0.398i)10-s + (0.854 − 0.519i)12-s + (−0.824 + 0.565i)13-s + (−0.662 + 0.749i)14-s + (−0.868 + 0.496i)15-s + (−0.894 − 0.447i)16-s + (0.531 + 0.847i)17-s + (−0.792 + 0.609i)18-s + ⋯
L(s)  = 1  + (0.620 + 0.784i)2-s + (−0.702 − 0.711i)3-s + (−0.230 + 0.973i)4-s + (0.256 − 0.966i)5-s + (0.122 − 0.992i)6-s + (0.176 + 0.984i)7-s + (−0.905 + 0.423i)8-s + (−0.0136 + 0.999i)9-s + (0.917 − 0.398i)10-s + (0.854 − 0.519i)12-s + (−0.824 + 0.565i)13-s + (−0.662 + 0.749i)14-s + (−0.868 + 0.496i)15-s + (−0.894 − 0.447i)16-s + (0.531 + 0.847i)17-s + (−0.792 + 0.609i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.880 - 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08861178245 + 0.3522041249i\)
\(L(\frac12)\) \(\approx\) \(-0.08861178245 + 0.3522041249i\)
\(L(1)\) \(\approx\) \(0.9107535326 + 0.3169360075i\)
\(L(1)\) \(\approx\) \(0.9107535326 + 0.3169360075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.620 + 0.784i)T \)
3 \( 1 + (-0.702 - 0.711i)T \)
5 \( 1 + (0.256 - 0.966i)T \)
7 \( 1 + (0.176 + 0.984i)T \)
13 \( 1 + (-0.824 + 0.565i)T \)
17 \( 1 + (0.531 + 0.847i)T \)
19 \( 1 + (0.839 - 0.542i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (-0.792 + 0.609i)T \)
31 \( 1 + (0.149 - 0.988i)T \)
37 \( 1 + (-0.662 - 0.749i)T \)
41 \( 1 + (0.484 + 0.874i)T \)
43 \( 1 + (-0.854 - 0.519i)T \)
53 \( 1 + (-0.981 + 0.190i)T \)
59 \( 1 + (0.758 + 0.652i)T \)
61 \( 1 + (-0.976 + 0.216i)T \)
67 \( 1 + (-0.990 + 0.136i)T \)
71 \( 1 + (-0.620 + 0.784i)T \)
73 \( 1 + (0.955 + 0.295i)T \)
79 \( 1 + (-0.410 + 0.911i)T \)
83 \( 1 + (-0.641 - 0.767i)T \)
89 \( 1 + (-0.775 + 0.631i)T \)
97 \( 1 + (-0.894 + 0.447i)T \)
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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

−22.68824424059712396431267817984, −22.23129934048730434621283276110, −21.14144035303633496592604602897, −20.662518157027596912450752171134, −19.677980506200219662487011577990, −18.66221864926055256520322883588, −17.80799529547544319087648020961, −17.04654676018006733950052823179, −15.86984630872563813788366717991, −14.927829972163496215969535655608, −14.28677096423658154150214012126, −13.456098969254675438864914912243, −12.24173601563181603688461796486, −11.44363912967631526019118951799, −10.69360464589776277895146966052, −10.03846974715098489142284664373, −9.4503937711522100784553881513, −7.498174435910127586552773053238, −6.61208577210190061653367361772, −5.46204479963572243925957552398, −4.81052481734599476166055460923, −3.5630253532103540943801122603, −2.99403474522749733945060299482, −1.3437076991535830894704409943, −0.08175400179874526052007201488, 1.53575248219925543517704574255, 2.71235138396558818994941248032, 4.39781824771504656702053313170, 5.29478962175363461087539125075, 5.72746648340840432524617329357, 6.845720581833485164005890280480, 7.768365439677027318719683119640, 8.679283819914233188046316527931, 9.54868953441449791611251188167, 11.28484559553988289330065603207, 12.08791267736116108907926936218, 12.64977061798918057742371692754, 13.35051952217867408518406442184, 14.40127304949981339762501010309, 15.32566587720491880356382656960, 16.37897577536643641103880218484, 16.91522313775685902416201364582, 17.65467532553449996879672882390, 18.51990428385662099247863629148, 19.49190891658132969357836472899, 20.77677528290854328955880437524, 21.6507518254502052420413336276, 22.16856085280399280900206104018, 23.16285178397040843516425758883, 24.15882456268239888019130665576

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