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Properties

Label 1-517-517.75-r0-0-0
Degree $1$
Conductor $517$
Sign $0.00648 + 0.999i$
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  + (0.256 + 0.966i)2-s + (−0.662 + 0.749i)3-s + (−0.868 + 0.496i)4-s + (0.721 − 0.692i)5-s + (−0.894 − 0.447i)6-s + (−0.999 − 0.0273i)7-s + (−0.702 − 0.711i)8-s + (−0.122 − 0.992i)9-s + (0.854 + 0.519i)10-s + (0.203 − 0.979i)12-s + (0.641 − 0.767i)13-s + (−0.230 − 0.973i)14-s + (0.0409 + 0.999i)15-s + (0.507 − 0.861i)16-s + (0.946 + 0.321i)17-s + (0.927 − 0.373i)18-s + ⋯
L(s)  = 1  + (0.256 + 0.966i)2-s + (−0.662 + 0.749i)3-s + (−0.868 + 0.496i)4-s + (0.721 − 0.692i)5-s + (−0.894 − 0.447i)6-s + (−0.999 − 0.0273i)7-s + (−0.702 − 0.711i)8-s + (−0.122 − 0.992i)9-s + (0.854 + 0.519i)10-s + (0.203 − 0.979i)12-s + (0.641 − 0.767i)13-s + (−0.230 − 0.973i)14-s + (0.0409 + 0.999i)15-s + (0.507 − 0.861i)16-s + (0.946 + 0.321i)17-s + (0.927 − 0.373i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00648 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00648 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.00648 + 0.999i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ 0.00648 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7799869914 + 0.7850643614i\)
\(L(\frac12)\) \(\approx\) \(0.7799869914 + 0.7850643614i\)
\(L(1)\) \(\approx\) \(0.7854672521 + 0.5247988658i\)
\(L(1)\) \(\approx\) \(0.7854672521 + 0.5247988658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.256 + 0.966i)T \)
3 \( 1 + (-0.662 + 0.749i)T \)
5 \( 1 + (0.721 - 0.692i)T \)
7 \( 1 + (-0.999 - 0.0273i)T \)
13 \( 1 + (0.641 - 0.767i)T \)
17 \( 1 + (0.946 + 0.321i)T \)
19 \( 1 + (-0.435 + 0.900i)T \)
23 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (0.927 - 0.373i)T \)
31 \( 1 + (0.976 + 0.216i)T \)
37 \( 1 + (-0.230 + 0.973i)T \)
41 \( 1 + (0.986 - 0.163i)T \)
43 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (0.149 - 0.988i)T \)
59 \( 1 + (0.994 - 0.109i)T \)
61 \( 1 + (-0.385 + 0.922i)T \)
67 \( 1 + (-0.334 - 0.942i)T \)
71 \( 1 + (0.256 - 0.966i)T \)
73 \( 1 + (0.905 + 0.423i)T \)
79 \( 1 + (-0.620 - 0.784i)T \)
83 \( 1 + (-0.0136 - 0.999i)T \)
89 \( 1 + (-0.990 + 0.136i)T \)
97 \( 1 + (0.507 + 0.861i)T \)
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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

−23.112006498969587289349798625431, −22.49833058201979351048185161676, −21.76103286417828344629913705085, −21.04588991810872981985952549123, −19.73909652269524882776553011995, −19.04486939905034684975254200392, −18.47632880022945210248211046440, −17.722721236487864088552120430572, −16.80321913442741652756942167557, −15.71227304735058292993748546473, −14.175892153997175253724991330172, −13.80763304392055828305969258828, −12.86191753962333188991661182361, −12.17375587643209759520125991615, −11.20401008472948605678858995814, −10.43365554646906906386335776915, −9.668119160307680085451488923801, −8.590776906190732589461345179381, −7.042613306017119134357486323305, −6.253282225205761089837249410660, −5.55723884849241108395845037973, −4.21846481034831777620764989282, −2.90724889036837628222559318668, −2.17068467143678811408756949020, −0.87006674816541775175008332231, 0.90779373203853328983494566013, 3.170175609680039367208916033041, 4.06560592206190107475953130392, 5.12679492520938806625098835954, 6.06028551553118915652910352909, 6.27075933704062984045690013310, 7.935302380989749963146905280381, 8.83160379535941209887970855621, 9.92957019907592416052292601915, 10.172791003856632863393850902575, 11.99095279316229927487440318955, 12.65459117429492143684592196666, 13.49430296785673676210547284275, 14.45751960718562273835585520050, 15.57074754639404631146377780051, 16.12786057363449063339160036588, 16.79118357468685813441214521470, 17.514110880967588435067502222443, 18.27892275414015870439544627180, 19.53562026133290712288820397002, 20.91829791148014547511411168927, 21.28631697475221232550316200515, 22.33007166412230444103689224161, 22.938715298293659981403100467810, 23.59029082442189136555294265564

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