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Properties

Label 1-517-517.128-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.312 + 0.949i$
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  + (0.999 + 0.0273i)2-s + (−0.740 + 0.672i)3-s + (0.998 + 0.0546i)4-s + (−0.969 − 0.243i)5-s + (−0.758 + 0.652i)6-s + (−0.946 − 0.321i)7-s + (0.996 + 0.0818i)8-s + (0.0954 − 0.995i)9-s + (−0.962 − 0.269i)10-s + (−0.775 + 0.631i)12-s + (0.484 − 0.874i)13-s + (−0.937 − 0.347i)14-s + (0.881 − 0.472i)15-s + (0.994 + 0.109i)16-s + (0.702 + 0.711i)17-s + (0.122 − 0.992i)18-s + ⋯
L(s)  = 1  + (0.999 + 0.0273i)2-s + (−0.740 + 0.672i)3-s + (0.998 + 0.0546i)4-s + (−0.969 − 0.243i)5-s + (−0.758 + 0.652i)6-s + (−0.946 − 0.321i)7-s + (0.996 + 0.0818i)8-s + (0.0954 − 0.995i)9-s + (−0.962 − 0.269i)10-s + (−0.775 + 0.631i)12-s + (0.484 − 0.874i)13-s + (−0.937 − 0.347i)14-s + (0.881 − 0.472i)15-s + (0.994 + 0.109i)16-s + (0.702 + 0.711i)17-s + (0.122 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.312 + 0.949i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8573728224 + 1.184916948i\)
\(L(\frac12)\) \(\approx\) \(0.8573728224 + 1.184916948i\)
\(L(1)\) \(\approx\) \(1.134147207 + 0.2461957995i\)
\(L(1)\) \(\approx\) \(1.134147207 + 0.2461957995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.999 + 0.0273i)T \)
3 \( 1 + (-0.740 + 0.672i)T \)
5 \( 1 + (-0.969 - 0.243i)T \)
7 \( 1 + (-0.946 - 0.321i)T \)
13 \( 1 + (0.484 - 0.874i)T \)
17 \( 1 + (0.702 + 0.711i)T \)
19 \( 1 + (-0.641 + 0.767i)T \)
23 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (0.122 - 0.992i)T \)
31 \( 1 + (-0.868 + 0.496i)T \)
37 \( 1 + (-0.937 + 0.347i)T \)
41 \( 1 + (0.385 + 0.922i)T \)
43 \( 1 + (0.775 + 0.631i)T \)
53 \( 1 + (-0.230 + 0.973i)T \)
59 \( 1 + (0.256 - 0.966i)T \)
61 \( 1 + (-0.0409 + 0.999i)T \)
67 \( 1 + (-0.576 + 0.816i)T \)
71 \( 1 + (-0.999 + 0.0273i)T \)
73 \( 1 + (-0.507 + 0.861i)T \)
79 \( 1 + (0.176 + 0.984i)T \)
83 \( 1 + (-0.986 + 0.163i)T \)
89 \( 1 + (-0.0682 - 0.997i)T \)
97 \( 1 + (0.994 - 0.109i)T \)
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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

−23.19121873794255043796075916404, −22.3859659585554122148110518339, −21.82555706408246575420995120553, −20.66191230990396139703691347088, −19.37900204501495786635274657810, −19.26614772819494526872375044802, −18.18205677631318639287431113886, −16.80482982324686362586537703623, −16.106963558341571209226606933043, −15.58528222370324275860197229958, −14.3624905330386939857360581842, −13.481707803517712359960221836205, −12.60510413572361753092981893875, −11.97723617594926938066730046472, −11.281431413041680017349789722861, −10.46050109767621932070842505489, −8.94972696275965257292168496628, −7.47089846935379200685482282596, −6.99430568753995693679072210434, −6.09555443432848652496155286097, −5.16612300729008294508298346415, −4.017053174972081425880727508684, −3.09062105528236130694185560614, −1.8633216212998004027471865023, −0.331604517776078916582623392439, 1.02149528311148111593750189812, 3.053697045175551769354194239846, 3.81701695487548636032494964173, 4.45234807890055014116788749643, 5.71295691083928118137764213856, 6.30440077326258028634364000897, 7.44822508615344527501247783248, 8.51594892864216418664644246637, 10.11160035164354347932761945372, 10.615599978364309882433766679, 11.597577584922292039971549331507, 12.579711577012100942059587263165, 12.82563863043624422213334148222, 14.36735615574531165771212444944, 15.1811618922993331702729529959, 15.93049124080854196043738048154, 16.454558290752429827485183795937, 17.21917323281066444411601879201, 18.73171723133015177349316822078, 19.64589278882487296506085383316, 20.47691788502530892312866012118, 21.12494762844712712113677580052, 22.20883546569224243867673284914, 22.88806104811342906569241816766, 23.25032335492358669644053950977

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