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Properties

Label 1-107-107.30-r0-0-0
Degree $1$
Conductor $107$
Sign $0.615 + 0.788i$
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  + (0.757 + 0.652i)2-s + (0.889 − 0.456i)3-s + (0.147 + 0.989i)4-s + (−0.430 + 0.902i)5-s + (0.972 + 0.234i)6-s + (0.674 − 0.737i)7-s + (−0.533 + 0.845i)8-s + (0.582 − 0.812i)9-s + (−0.915 + 0.403i)10-s + (−0.998 + 0.0592i)11-s + (0.582 + 0.812i)12-s + (−0.861 − 0.508i)13-s + (0.992 − 0.118i)14-s + (0.0296 + 0.999i)15-s + (−0.956 + 0.292i)16-s + (−0.205 + 0.978i)17-s + ⋯
L(s)  = 1  + (0.757 + 0.652i)2-s + (0.889 − 0.456i)3-s + (0.147 + 0.989i)4-s + (−0.430 + 0.902i)5-s + (0.972 + 0.234i)6-s + (0.674 − 0.737i)7-s + (−0.533 + 0.845i)8-s + (0.582 − 0.812i)9-s + (−0.915 + 0.403i)10-s + (−0.998 + 0.0592i)11-s + (0.582 + 0.812i)12-s + (−0.861 − 0.508i)13-s + (0.992 − 0.118i)14-s + (0.0296 + 0.999i)15-s + (−0.956 + 0.292i)16-s + (−0.205 + 0.978i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.621441011 + 0.7912076095i\)
\(L(\frac12)\) \(\approx\) \(1.621441011 + 0.7912076095i\)
\(L(1)\) \(\approx\) \(1.640514314 + 0.5699170984i\)
\(L(1)\) \(\approx\) \(1.640514314 + 0.5699170984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.757 + 0.652i)T \)
3 \( 1 + (0.889 - 0.456i)T \)
5 \( 1 + (-0.430 + 0.902i)T \)
7 \( 1 + (0.674 - 0.737i)T \)
11 \( 1 + (-0.998 + 0.0592i)T \)
13 \( 1 + (-0.861 - 0.508i)T \)
17 \( 1 + (-0.205 + 0.978i)T \)
19 \( 1 + (0.482 - 0.875i)T \)
23 \( 1 + (0.992 + 0.118i)T \)
29 \( 1 + (-0.717 - 0.696i)T \)
31 \( 1 + (0.937 + 0.348i)T \)
37 \( 1 + (-0.320 + 0.947i)T \)
41 \( 1 + (-0.984 - 0.176i)T \)
43 \( 1 + (-0.430 - 0.902i)T \)
47 \( 1 + (-0.984 + 0.176i)T \)
53 \( 1 + (0.757 - 0.652i)T \)
59 \( 1 + (-0.717 + 0.696i)T \)
61 \( 1 + (0.674 + 0.737i)T \)
67 \( 1 + (-0.533 - 0.845i)T \)
71 \( 1 + (0.889 + 0.456i)T \)
73 \( 1 + (-0.794 + 0.606i)T \)
79 \( 1 + (0.829 + 0.558i)T \)
83 \( 1 + (-0.0887 + 0.996i)T \)
89 \( 1 + (0.972 - 0.234i)T \)
97 \( 1 + (-0.915 + 0.403i)T \)
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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

−29.5011421834874900778665985257, −28.481725524313840354137291356617, −27.5575229076483248648170096118, −26.68223177116778500584673355096, −24.808397250417031840211068623802, −24.57117655269351282654132087670, −23.25597320279354288999797387500, −21.89821748065179240837398664497, −20.94679480767425203186589037488, −20.51223350769894837507079242340, −19.331382977606984600652817967502, −18.41283664632844622868252519132, −16.366480819491884513262917636511, −15.41469388140973433879566855447, −14.543794127198980943732184772272, −13.39795465589308673768095999858, −12.35247906056613435563564732381, −11.28919390341795905608864041034, −9.82356234077526105066468453843, −8.84788977710955849755844329223, −7.56739603884720090636501769010, −5.211823014330039059114132549712, −4.68261298045441538056949482770, −3.11534004083854830151139328042, −1.879355403325189162467610606516, 2.469538655415903299859839154379, 3.54049182139200141517827652870, 4.92123504042470233629164912359, 6.77846855273026119677780429231, 7.54610432991544210852736116020, 8.33160996063833714235976001392, 10.308854804441104548797050812550, 11.67501739448583515941266074580, 13.0644387828279200540092565447, 13.83147559272344811347481132886, 15.04128093484321974156339238819, 15.320802957220134996672486009881, 17.199734093392084492642244467786, 18.122348831479498534662289756754, 19.45255889359313991037178346524, 20.525411154932364073587630750104, 21.52247049068316161357856704600, 22.79527256159157890510802029324, 23.79314935685449685478646268802, 24.3997441069502676178076878089, 25.73468355891389993773109053226, 26.46550733440757232374837918656, 27.089901719274953283408801451180, 29.25157779823616901432807847301, 30.3830641984568575970005193074

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