↑

Properties

Label 1-107-107.34-r0-0-0
Degree $1$
Conductor $107$
Sign $0.214 + 0.976i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.205 + 0.978i)2-s + (0.375 − 0.926i)3-s + (−0.915 − 0.403i)4-s + (−0.320 + 0.947i)5-s + (0.829 + 0.558i)6-s + (0.482 + 0.875i)7-s + (0.582 − 0.812i)8-s + (−0.717 − 0.696i)9-s + (−0.861 − 0.508i)10-s + (0.147 + 0.989i)11-s + (−0.717 + 0.696i)12-s + (0.972 − 0.234i)13-s + (−0.956 + 0.292i)14-s + (0.757 + 0.652i)15-s + (0.674 + 0.737i)16-s + (0.263 + 0.964i)17-s + ⋯
L(s)  = 1  + (−0.205 + 0.978i)2-s + (0.375 − 0.926i)3-s + (−0.915 − 0.403i)4-s + (−0.320 + 0.947i)5-s + (0.829 + 0.558i)6-s + (0.482 + 0.875i)7-s + (0.582 − 0.812i)8-s + (−0.717 − 0.696i)9-s + (−0.861 − 0.508i)10-s + (0.147 + 0.989i)11-s + (−0.717 + 0.696i)12-s + (0.972 − 0.234i)13-s + (−0.956 + 0.292i)14-s + (0.757 + 0.652i)15-s + (0.674 + 0.737i)16-s + (0.263 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.214 + 0.976i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.214 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7366971883 + 0.5927156400i\)
\(L(\frac12)\) \(\approx\) \(0.7366971883 + 0.5927156400i\)
\(L(1)\) \(\approx\) \(0.8810384648 + 0.4125738017i\)
\(L(1)\) \(\approx\) \(0.8810384648 + 0.4125738017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.205 + 0.978i)T \)
3 \( 1 + (0.375 - 0.926i)T \)
5 \( 1 + (-0.320 + 0.947i)T \)
7 \( 1 + (0.482 + 0.875i)T \)
11 \( 1 + (0.147 + 0.989i)T \)
13 \( 1 + (0.972 - 0.234i)T \)
17 \( 1 + (0.263 + 0.964i)T \)
19 \( 1 + (0.889 + 0.456i)T \)
23 \( 1 + (-0.956 - 0.292i)T \)
29 \( 1 + (0.937 + 0.348i)T \)
31 \( 1 + (-0.630 - 0.776i)T \)
37 \( 1 + (0.0296 - 0.999i)T \)
41 \( 1 + (-0.430 + 0.902i)T \)
43 \( 1 + (-0.320 - 0.947i)T \)
47 \( 1 + (-0.430 - 0.902i)T \)
53 \( 1 + (-0.205 - 0.978i)T \)
59 \( 1 + (0.937 - 0.348i)T \)
61 \( 1 + (0.482 - 0.875i)T \)
67 \( 1 + (0.582 + 0.812i)T \)
71 \( 1 + (0.375 + 0.926i)T \)
73 \( 1 + (-0.998 + 0.0592i)T \)
79 \( 1 + (-0.0887 - 0.996i)T \)
83 \( 1 + (-0.533 - 0.845i)T \)
89 \( 1 + (0.829 - 0.558i)T \)
97 \( 1 + (-0.861 - 0.508i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.23233048208496993160181893149, −28.2787258193247506311234861383, −27.30523297515906778993787561618, −26.86154686908424178702098251892, −25.648048499573187440432191805072, −24.09430492984366492983386663757, −23.02075819954568490138278265056, −21.75887478363727067139381236701, −20.84300999594425178081297264985, −20.27162698871825784102918454651, −19.397046113544633569509255782324, −17.90052590549547358793035717873, −16.63405192658743205296076303292, −15.97035201701014660489020681595, −13.98939619204946245530974078827, −13.58520749765171191823057054644, −11.78089918653028025273278138225, −11.034247256365486374844609402179, −9.80010650987821342179373854211, −8.76088788421971068657298113206, −7.93456853867875047391770024617, −5.245074929081873750473756825742, −4.215210351250909003197580534655, −3.242937195476822221182091208729, −1.14161044345418275754773694474, 1.85572636402622355755151751814, 3.6924234554893657264858775584, 5.67566374219108233881054951173, 6.65006969542382298130705162203, 7.77687523681528534684790692171, 8.54155159834462616752295554254, 10.03844749923105978222083889639, 11.691405531653565737055043013078, 12.88588460099981523154800893951, 14.276130613555077975319879935879, 14.82925537027830107139141212451, 15.871126552941980522663005125058, 17.64224570760686453458867187420, 18.22092114126882002525561989496, 18.93483748658565187312284788757, 20.17829485187209349224263308652, 21.92560778462736811748451367585, 23.012689348133215779581840680399, 23.74225597315350467312003602904, 24.91355217301318363844205555572, 25.61009589574817553910883965824, 26.39515236349373590620860223274, 27.720426240989067686921728488255, 28.57098941613432805212817121186, 30.326104044384347751933312443027

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