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Properties

Label 1-517-517.61-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.968 - 0.247i$
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  + (−0.976 + 0.216i)2-s + (0.927 − 0.373i)3-s + (0.905 − 0.423i)4-s + (−0.385 − 0.922i)5-s + (−0.824 + 0.565i)6-s + (0.868 + 0.496i)7-s + (−0.792 + 0.609i)8-s + (0.721 − 0.692i)9-s + (0.576 + 0.816i)10-s + (0.682 − 0.730i)12-s + (0.620 − 0.784i)13-s + (−0.955 − 0.295i)14-s + (−0.702 − 0.711i)15-s + (0.641 − 0.767i)16-s + (−0.998 + 0.0546i)17-s + (−0.554 + 0.832i)18-s + ⋯
L(s)  = 1  + (−0.976 + 0.216i)2-s + (0.927 − 0.373i)3-s + (0.905 − 0.423i)4-s + (−0.385 − 0.922i)5-s + (−0.824 + 0.565i)6-s + (0.868 + 0.496i)7-s + (−0.792 + 0.609i)8-s + (0.721 − 0.692i)9-s + (0.576 + 0.816i)10-s + (0.682 − 0.730i)12-s + (0.620 − 0.784i)13-s + (−0.955 − 0.295i)14-s + (−0.702 − 0.711i)15-s + (0.641 − 0.767i)16-s + (−0.998 + 0.0546i)17-s + (−0.554 + 0.832i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.968 - 0.247i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.968 - 0.247i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1133270473 - 0.9003136129i\)
\(L(\frac12)\) \(\approx\) \(0.1133270473 - 0.9003136129i\)
\(L(1)\) \(\approx\) \(0.8025240259 - 0.2786064580i\)
\(L(1)\) \(\approx\) \(0.8025240259 - 0.2786064580i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.976 + 0.216i)T \)
3 \( 1 + (0.927 - 0.373i)T \)
5 \( 1 + (-0.385 - 0.922i)T \)
7 \( 1 + (0.868 + 0.496i)T \)
13 \( 1 + (0.620 - 0.784i)T \)
17 \( 1 + (-0.998 + 0.0546i)T \)
19 \( 1 + (-0.758 - 0.652i)T \)
23 \( 1 + (-0.917 + 0.398i)T \)
29 \( 1 + (-0.554 + 0.832i)T \)
31 \( 1 + (-0.531 - 0.847i)T \)
37 \( 1 + (-0.955 + 0.295i)T \)
41 \( 1 + (0.999 + 0.0273i)T \)
43 \( 1 + (-0.682 - 0.730i)T \)
53 \( 1 + (-0.282 - 0.959i)T \)
59 \( 1 + (-0.484 - 0.874i)T \)
61 \( 1 + (-0.946 + 0.321i)T \)
67 \( 1 + (0.203 + 0.979i)T \)
71 \( 1 + (0.976 + 0.216i)T \)
73 \( 1 + (0.435 - 0.900i)T \)
79 \( 1 + (-0.149 - 0.988i)T \)
83 \( 1 + (-0.256 - 0.966i)T \)
89 \( 1 + (0.854 + 0.519i)T \)
97 \( 1 + (0.641 + 0.767i)T \)
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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

−24.03385593031265030650220060002, −22.84066687281538933061585529372, −21.56887440285710072099471410263, −21.14625182955153998569579273359, −20.13742266571438969014009281301, −19.592969797922078698677397328959, −18.62650930790213847575291559281, −18.13566086406642503624518635358, −16.99693306059960549992950345858, −16.020283222899702165348969543630, −15.27901123090730439468824789877, −14.434340374742054371938577330242, −13.70106306380405751716844571211, −12.28354390750402457194294960140, −11.04621899797877594979142120934, −10.781213770602394146305696578594, −9.740839235211207113978878210519, −8.70063138756055305924082734896, −8.02965738939507318208940536250, −7.23126522870570952113929019357, −6.307998430307677839129402132580, −4.30457516116853683559063857969, −3.66057117585389486119260799889, −2.369646291385749921224256561988, −1.64904677316214264900770136223, 0.256586812218552365851431019040, 1.51791489009861040708869424885, 2.2401455649905788275538773932, 3.682904286345686191781543094526, 4.99943520340400017307841288792, 6.16522316627794259886016855184, 7.403410685201296232769892756612, 8.16431849166556784769159486798, 8.7215931138051250506638409371, 9.34320277098559482835714413456, 10.68721535189408094143885561569, 11.594932753357192233372644624248, 12.56929368325430274950133237557, 13.45581039366351330444769046683, 14.697632174286915554377003627409, 15.40852676417342398118981293269, 15.96456840462970360894330981368, 17.27993823189109707420731686077, 17.91167894230218956204125673007, 18.7092041068719672331391310793, 19.65091962614626681284528361546, 20.289832053445180280528628406478, 20.79408301043205270907939257618, 21.78797747756420255564875216213, 23.51470089295991805338735529945

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