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Properties

Label 1-107-107.12-r0-0-0
Degree $1$
Conductor $107$
Sign $0.332 - 0.943i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
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  + (−0.430 − 0.902i)2-s + (−0.956 − 0.292i)3-s + (−0.630 + 0.776i)4-s + (0.889 − 0.456i)5-s + (0.147 + 0.989i)6-s + (0.263 + 0.964i)7-s + (0.972 + 0.234i)8-s + (0.829 + 0.558i)9-s + (−0.794 − 0.606i)10-s + (0.937 − 0.348i)11-s + (0.829 − 0.558i)12-s + (−0.998 − 0.0592i)13-s + (0.757 − 0.652i)14-s + (−0.984 + 0.176i)15-s + (−0.205 − 0.978i)16-s + (−0.320 − 0.947i)17-s + ⋯
L(s)  = 1  + (−0.430 − 0.902i)2-s + (−0.956 − 0.292i)3-s + (−0.630 + 0.776i)4-s + (0.889 − 0.456i)5-s + (0.147 + 0.989i)6-s + (0.263 + 0.964i)7-s + (0.972 + 0.234i)8-s + (0.829 + 0.558i)9-s + (−0.794 − 0.606i)10-s + (0.937 − 0.348i)11-s + (0.829 − 0.558i)12-s + (−0.998 − 0.0592i)13-s + (0.757 − 0.652i)14-s + (−0.984 + 0.176i)15-s + (−0.205 − 0.978i)16-s + (−0.320 − 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.332 - 0.943i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.332 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5878487496 - 0.4161656256i\)
\(L(\frac12)\) \(\approx\) \(0.5878487496 - 0.4161656256i\)
\(L(1)\) \(\approx\) \(0.6798902140 - 0.3335542338i\)
\(L(1)\) \(\approx\) \(0.6798902140 - 0.3335542338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.430 - 0.902i)T \)
3 \( 1 + (-0.956 - 0.292i)T \)
5 \( 1 + (0.889 - 0.456i)T \)
7 \( 1 + (0.263 + 0.964i)T \)
11 \( 1 + (0.937 - 0.348i)T \)
13 \( 1 + (-0.998 - 0.0592i)T \)
17 \( 1 + (-0.320 - 0.947i)T \)
19 \( 1 + (0.992 - 0.118i)T \)
23 \( 1 + (0.757 + 0.652i)T \)
29 \( 1 + (-0.0887 - 0.996i)T \)
31 \( 1 + (-0.533 + 0.845i)T \)
37 \( 1 + (0.375 - 0.926i)T \)
41 \( 1 + (0.482 + 0.875i)T \)
43 \( 1 + (0.889 + 0.456i)T \)
47 \( 1 + (0.482 - 0.875i)T \)
53 \( 1 + (-0.430 + 0.902i)T \)
59 \( 1 + (-0.0887 + 0.996i)T \)
61 \( 1 + (0.263 - 0.964i)T \)
67 \( 1 + (0.972 - 0.234i)T \)
71 \( 1 + (-0.956 + 0.292i)T \)
73 \( 1 + (-0.717 + 0.696i)T \)
79 \( 1 + (-0.915 - 0.403i)T \)
83 \( 1 + (-0.861 - 0.508i)T \)
89 \( 1 + (0.147 - 0.989i)T \)
97 \( 1 + (-0.794 - 0.606i)T \)
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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

−29.48557324700791613763025686509, −28.78095132889219527044784787668, −27.49004618134333557107388055615, −26.82062628064702416185949807463, −25.868109851687619267429789441212, −24.57517902019308549315595334105, −23.82710899811254714669089400020, −22.48758172520201883112089039092, −22.12904632109889323591259380089, −20.46701411451596537739413076623, −19.026369979253085300585902708359, −17.73227292142372023710843417990, −17.23714071365070902494443318869, −16.511952551114600108996400736823, −14.95310057104193295014633682393, −14.204894518821306369699022322845, −12.81801229217351456872542899055, −11.05886715044781640314920837995, −10.14260947276003265487807371256, −9.29421577630795950293624576669, −7.3112633676988767628715986996, −6.59159697451269578673221292778, −5.38388673930203729829917920808, −4.21394862948259348157115491145, −1.36280321863161345966842563105, 1.22127061519362479953674042217, 2.542189088152522593996921856774, 4.71560249562422398606003842566, 5.68672382553171341715455645961, 7.33591343756439927517888812127, 9.04925515038774609844603782591, 9.73910232027026015519473556058, 11.26954814920514185843388269364, 11.9964011113020380332518523281, 12.94729597214855433555859585250, 14.14064835073313950953136110453, 16.1217104237720961311117034688, 17.22123004373885130387820525428, 17.83667014437612584028423809337, 18.80649433648249338681454454503, 19.9891991041294337254249965403, 21.4240236521615277855604722999, 21.894323614615946550611811819, 22.825222852006284550698844539417, 24.64401516484030613085861142076, 24.98298490766237596022746336011, 26.809868837127558901314326929246, 27.671132087700510720055156173741, 28.59981614413501094042285236267, 29.23461742603054088560607315868

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