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Properties

Label 1-517-517.27-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.726 - 0.687i$
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  + (0.721 − 0.692i)2-s + (−0.230 + 0.973i)3-s + (0.0409 − 0.999i)4-s + (0.824 − 0.565i)5-s + (0.507 + 0.861i)6-s + (−0.969 − 0.243i)7-s + (−0.662 − 0.749i)8-s + (−0.894 − 0.447i)9-s + (0.203 − 0.979i)10-s + (0.962 + 0.269i)12-s + (−0.0136 − 0.999i)13-s + (−0.868 + 0.496i)14-s + (0.360 + 0.932i)15-s + (−0.996 − 0.0818i)16-s + (−0.981 + 0.190i)17-s + (−0.955 + 0.295i)18-s + ⋯
L(s)  = 1  + (0.721 − 0.692i)2-s + (−0.230 + 0.973i)3-s + (0.0409 − 0.999i)4-s + (0.824 − 0.565i)5-s + (0.507 + 0.861i)6-s + (−0.969 − 0.243i)7-s + (−0.662 − 0.749i)8-s + (−0.894 − 0.447i)9-s + (0.203 − 0.979i)10-s + (0.962 + 0.269i)12-s + (−0.0136 − 0.999i)13-s + (−0.868 + 0.496i)14-s + (0.360 + 0.932i)15-s + (−0.996 − 0.0818i)16-s + (−0.981 + 0.190i)17-s + (−0.955 + 0.295i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4835489178 - 1.214015623i\)
\(L(\frac12)\) \(\approx\) \(0.4835489178 - 1.214015623i\)
\(L(1)\) \(\approx\) \(1.068742839 - 0.5827568850i\)
\(L(1)\) \(\approx\) \(1.068742839 - 0.5827568850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.721 - 0.692i)T \)
3 \( 1 + (-0.230 + 0.973i)T \)
5 \( 1 + (0.824 - 0.565i)T \)
7 \( 1 + (-0.969 - 0.243i)T \)
13 \( 1 + (-0.0136 - 0.999i)T \)
17 \( 1 + (-0.981 + 0.190i)T \)
19 \( 1 + (0.792 - 0.609i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.955 + 0.295i)T \)
31 \( 1 + (-0.385 + 0.922i)T \)
37 \( 1 + (-0.868 - 0.496i)T \)
41 \( 1 + (0.0954 - 0.995i)T \)
43 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (0.976 - 0.216i)T \)
59 \( 1 + (0.554 - 0.832i)T \)
61 \( 1 + (0.410 - 0.911i)T \)
67 \( 1 + (-0.0682 + 0.997i)T \)
71 \( 1 + (0.721 + 0.692i)T \)
73 \( 1 + (-0.702 - 0.711i)T \)
79 \( 1 + (0.256 - 0.966i)T \)
83 \( 1 + (-0.122 - 0.992i)T \)
89 \( 1 + (-0.334 + 0.942i)T \)
97 \( 1 + (-0.996 + 0.0818i)T \)
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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

−24.12932010280805054783552557551, −22.82097725246478794041301417870, −22.48363046576712613505513176068, −21.77692399534031463700298732522, −20.66067767070190354821182826996, −19.55716979174518390719663924332, −18.49116623842608284652454085792, −18.02133051304036428370338652409, −16.941466168421019734131020146815, −16.35879809222672408797817898442, −15.23061575266201065163584460157, −14.16755832653509343108308767640, −13.6108569575338798475974222370, −12.9668743730206565485526729593, −11.98936652247833776961549377290, −11.22790209302976257813024119092, −9.73237587351176877670793280676, −8.834220236627696367173777886164, −7.53518811980066305155780597072, −6.80648591010899340838207542903, −6.11446121857922893897580478535, −5.49876724313620786125770211602, −3.95806493918200079110520298363, −2.75325556139780067163018270268, −1.953029612078201747262152782159, 0.51752773391582464866571931374, 2.19806542540343405569593483492, 3.274159550012215345800552449986, 4.12501768758570485518421493811, 5.272527836237121062942177611552, 5.753273273805198123799667839942, 6.838813529644687610318705802290, 8.810735716612869127216759027706, 9.47308886709899772128505784472, 10.278790952782752304857296332672, 10.85324638103089038996853634076, 12.10825289076357153854485755378, 12.899104518825264963964555441140, 13.6560126591862974514913750634, 14.52201518484769918363251592020, 15.7473105565059226086071799909, 16.02920045274936758727566248647, 17.34571517438791121288980549885, 18.026496608203808867550392465316, 19.471333759334538948136847593306, 20.28163036297059989229939002414, 20.58083739697322748278223044560, 21.84008454652692551635144500121, 22.15082547716504597585038247013, 22.83161561024715160256042328281

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