Properties

Label 1-517-517.17-r1-0-0
Degree $1$
Conductor $517$
Sign $0.341 - 0.939i$
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  + (0.531 + 0.847i)2-s + (0.554 + 0.832i)3-s + (−0.435 + 0.900i)4-s + (0.946 − 0.321i)5-s + (−0.410 + 0.911i)6-s + (−0.905 + 0.423i)7-s + (−0.994 + 0.109i)8-s + (−0.385 + 0.922i)9-s + (0.775 + 0.631i)10-s + (−0.990 + 0.136i)12-s + (−0.149 − 0.988i)13-s + (−0.839 − 0.542i)14-s + (0.792 + 0.609i)15-s + (−0.620 − 0.784i)16-s + (−0.507 − 0.861i)17-s + (−0.986 + 0.163i)18-s + ⋯
L(s)  = 1  + (0.531 + 0.847i)2-s + (0.554 + 0.832i)3-s + (−0.435 + 0.900i)4-s + (0.946 − 0.321i)5-s + (−0.410 + 0.911i)6-s + (−0.905 + 0.423i)7-s + (−0.994 + 0.109i)8-s + (−0.385 + 0.922i)9-s + (0.775 + 0.631i)10-s + (−0.990 + 0.136i)12-s + (−0.149 − 0.988i)13-s + (−0.839 − 0.542i)14-s + (0.792 + 0.609i)15-s + (−0.620 − 0.784i)16-s + (−0.507 − 0.861i)17-s + (−0.986 + 0.163i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.341 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $0.341 - 0.939i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ 0.341 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1792539770 + 0.1255958600i\)
\(L(\frac12)\) \(\approx\) \(-0.1792539770 + 0.1255958600i\)
\(L(1)\) \(\approx\) \(0.8891532962 + 0.8404977333i\)
\(L(1)\) \(\approx\) \(0.8891532962 + 0.8404977333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.531 + 0.847i)T \)
3 \( 1 + (0.554 + 0.832i)T \)
5 \( 1 + (0.946 - 0.321i)T \)
7 \( 1 + (-0.905 + 0.423i)T \)
13 \( 1 + (-0.149 - 0.988i)T \)
17 \( 1 + (-0.507 - 0.861i)T \)
19 \( 1 + (-0.598 + 0.800i)T \)
23 \( 1 + (-0.0682 - 0.997i)T \)
29 \( 1 + (-0.986 + 0.163i)T \)
31 \( 1 + (-0.937 + 0.347i)T \)
37 \( 1 + (-0.839 + 0.542i)T \)
41 \( 1 + (0.868 - 0.496i)T \)
43 \( 1 + (0.990 + 0.136i)T \)
53 \( 1 + (-0.740 - 0.672i)T \)
59 \( 1 + (-0.176 + 0.984i)T \)
61 \( 1 + (-0.998 + 0.0546i)T \)
67 \( 1 + (0.682 - 0.730i)T \)
71 \( 1 + (-0.531 + 0.847i)T \)
73 \( 1 + (-0.758 + 0.652i)T \)
79 \( 1 + (0.282 + 0.959i)T \)
83 \( 1 + (-0.976 - 0.216i)T \)
89 \( 1 + (-0.576 + 0.816i)T \)
97 \( 1 + (-0.620 + 0.784i)T \)
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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

−22.52482372815764170220438039047, −21.77570517897433387392829031767, −21.02271867087669206616347493095, −20.03654504430357816772294200879, −19.320976366735829911725322718061, −18.820171002460635513986163040813, −17.769751961129020702992476906411, −17.046440223172172667064522118167, −15.49544619319486541650576776470, −14.51126886200249641200476423672, −13.856398972066727070035990469854, −13.07941071854202537478633670248, −12.70252722627477210034629590307, −11.39712089299513423825358577011, −10.55239059287046719994404302301, −9.280626196031939285656895154903, −9.16988231134991333110290141741, −7.324013386297797323098819244223, −6.43515496277932085616621750181, −5.789474583334835060577528273961, −4.20140077724787200297430261140, −3.25762884586927730013702809836, −2.231907056362004853097230874997, −1.550833006971070750537424394943, −0.03713600158341612555050342551, 2.35489253403907274028790187282, 3.15730428516250171721809153834, 4.27562492000993574489587554384, 5.344707590239993860284402366553, 5.91132184563958486140839771494, 7.08192276947611226951109557759, 8.33351614851132664407202643540, 9.072589809379238132843528606046, 9.761101499808492759959725601831, 10.77232122144671235883581399720, 12.44831951191359206661232255565, 12.98218776753018497227225656912, 13.91266601073446577826871745826, 14.64617991233872900861366340799, 15.50883648158639176256438605861, 16.27337381000659021295970842502, 16.865629516614516661613688879409, 17.890649591245176708746287665797, 18.846076498865212470837507805, 20.21047582596112299089615689396, 20.795874118257313750034791160463, 21.66962566492851157445703143732, 22.49025875449496897167977387918, 22.73876743185849264808495062927, 24.4077250960633439073028657282

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