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Properties

Label 1-517-517.157-r0-0-0
Degree $1$
Conductor $517$
Sign $-0.750 - 0.661i$
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  + (0.986 − 0.163i)2-s + (−0.282 − 0.959i)3-s + (0.946 − 0.321i)4-s + (0.0954 − 0.995i)5-s + (−0.435 − 0.900i)6-s + (−0.385 − 0.922i)7-s + (0.881 − 0.472i)8-s + (−0.839 + 0.542i)9-s + (−0.0682 − 0.997i)10-s + (−0.576 − 0.816i)12-s + (0.994 + 0.109i)13-s + (−0.531 − 0.847i)14-s + (−0.981 + 0.190i)15-s + (0.792 − 0.609i)16-s + (0.0409 + 0.999i)17-s + (−0.740 + 0.672i)18-s + ⋯
L(s)  = 1  + (0.986 − 0.163i)2-s + (−0.282 − 0.959i)3-s + (0.946 − 0.321i)4-s + (0.0954 − 0.995i)5-s + (−0.435 − 0.900i)6-s + (−0.385 − 0.922i)7-s + (0.881 − 0.472i)8-s + (−0.839 + 0.542i)9-s + (−0.0682 − 0.997i)10-s + (−0.576 − 0.816i)12-s + (0.994 + 0.109i)13-s + (−0.531 − 0.847i)14-s + (−0.981 + 0.190i)15-s + (0.792 − 0.609i)16-s + (0.0409 + 0.999i)17-s + (−0.740 + 0.672i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.750 - 0.661i$
Analytic conductor: \(2.40093\)
Root analytic conductor: \(2.40093\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (0:\ ),\ -0.750 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8056782143 - 2.133374708i\)
\(L(\frac12)\) \(\approx\) \(0.8056782143 - 2.133374708i\)
\(L(1)\) \(\approx\) \(1.312717841 - 1.136922902i\)
\(L(1)\) \(\approx\) \(1.312717841 - 1.136922902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.986 - 0.163i)T \)
3 \( 1 + (-0.282 - 0.959i)T \)
5 \( 1 + (0.0954 - 0.995i)T \)
7 \( 1 + (-0.385 - 0.922i)T \)
13 \( 1 + (0.994 + 0.109i)T \)
17 \( 1 + (0.0409 + 0.999i)T \)
19 \( 1 + (0.507 - 0.861i)T \)
23 \( 1 + (0.460 - 0.887i)T \)
29 \( 1 + (-0.740 + 0.672i)T \)
31 \( 1 + (-0.999 + 0.0273i)T \)
37 \( 1 + (-0.531 + 0.847i)T \)
41 \( 1 + (0.721 - 0.692i)T \)
43 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (-0.176 + 0.984i)T \)
59 \( 1 + (-0.0136 + 0.999i)T \)
61 \( 1 + (-0.969 + 0.243i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (0.986 + 0.163i)T \)
73 \( 1 + (0.998 - 0.0546i)T \)
79 \( 1 + (-0.484 - 0.874i)T \)
83 \( 1 + (0.554 + 0.832i)T \)
89 \( 1 + (-0.917 - 0.398i)T \)
97 \( 1 + (0.792 + 0.609i)T \)
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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

−23.43887727263536657576908423859, −22.76483912556032260596274741664, −22.383546238421939018344361240688, −21.46711032522088507201843512071, −20.95590799455172923555158372103, −19.961524385546611678647628084, −18.77403916819998863349272902544, −17.950373942532568692470699234960, −16.73392456648478639377867544355, −15.82443270033820838846456517889, −15.44602909372198898283487215715, −14.53433420526282737855452999106, −13.82996274752782973530111199105, −12.68139207714082710593559519718, −11.500890766166354507401399678728, −11.22631533389975657118126219858, −10.06438827029731048594924651874, −9.17599997666005496144117065599, −7.80490348069802403787255906793, −6.63916316375933252581849322921, −5.77695669906789244779279315988, −5.24173353346287530768087389542, −3.68433145226497694777530103913, −3.31240129439837513453626002530, −2.13942058092913606704278007311, 0.94803780865890398974861951730, 1.75050713242013607436592452505, 3.20978756762265325221330978903, 4.261959032580057453016582855238, 5.28980123942652272783436676872, 6.19474494249982146241783760835, 6.99499617322468847918889076778, 7.971850982331421382515451499152, 9.10072379837146588566690257848, 10.58886190748036373980119335188, 11.21374137410556844652333183602, 12.35110734222164581847270047827, 12.96572223791548583747899660956, 13.478292937302259477699417983026, 14.27436354111012084360929572361, 15.558137053669025675605551656384, 16.60430429874737865411698857716, 16.94185866385756803598112123644, 18.20584058748664871319006103491, 19.3107058954239757548841767038, 20.0388911219533796252329367089, 20.56402339986696523906460318354, 21.62021214071505606907273265294, 22.64523647544447951502481280877, 23.32562778683747958937763963007

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