Properties

Label 1-107-107.11-r0-0-0
Degree $1$
Conductor $107$
Sign $0.999 + 0.0278i$
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  + (0.263 + 0.964i)2-s + (−0.984 − 0.176i)3-s + (−0.861 + 0.508i)4-s + (0.0296 − 0.999i)5-s + (−0.0887 − 0.996i)6-s + (0.889 − 0.456i)7-s + (−0.717 − 0.696i)8-s + (0.937 + 0.348i)9-s + (0.972 − 0.234i)10-s + (−0.915 − 0.403i)11-s + (0.937 − 0.348i)12-s + (0.829 − 0.558i)13-s + (0.674 + 0.737i)14-s + (−0.205 + 0.978i)15-s + (0.482 − 0.875i)16-s + (0.992 + 0.118i)17-s + ⋯
L(s)  = 1  + (0.263 + 0.964i)2-s + (−0.984 − 0.176i)3-s + (−0.861 + 0.508i)4-s + (0.0296 − 0.999i)5-s + (−0.0887 − 0.996i)6-s + (0.889 − 0.456i)7-s + (−0.717 − 0.696i)8-s + (0.937 + 0.348i)9-s + (0.972 − 0.234i)10-s + (−0.915 − 0.403i)11-s + (0.937 − 0.348i)12-s + (0.829 − 0.558i)13-s + (0.674 + 0.737i)14-s + (−0.205 + 0.978i)15-s + (0.482 − 0.875i)16-s + (0.992 + 0.118i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.999 + 0.0278i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.999 + 0.0278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8348328728 + 0.01164232054i\)
\(L(\frac12)\) \(\approx\) \(0.8348328728 + 0.01164232054i\)
\(L(1)\) \(\approx\) \(0.8738478650 + 0.1363201062i\)
\(L(1)\) \(\approx\) \(0.8738478650 + 0.1363201062i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.263 + 0.964i)T \)
3 \( 1 + (-0.984 - 0.176i)T \)
5 \( 1 + (0.0296 - 0.999i)T \)
7 \( 1 + (0.889 - 0.456i)T \)
11 \( 1 + (-0.915 - 0.403i)T \)
13 \( 1 + (0.829 - 0.558i)T \)
17 \( 1 + (0.992 + 0.118i)T \)
19 \( 1 + (0.375 + 0.926i)T \)
23 \( 1 + (0.674 - 0.737i)T \)
29 \( 1 + (-0.630 - 0.776i)T \)
31 \( 1 + (-0.794 - 0.606i)T \)
37 \( 1 + (0.757 - 0.652i)T \)
41 \( 1 + (-0.320 + 0.947i)T \)
43 \( 1 + (0.0296 + 0.999i)T \)
47 \( 1 + (-0.320 - 0.947i)T \)
53 \( 1 + (0.263 - 0.964i)T \)
59 \( 1 + (-0.630 + 0.776i)T \)
61 \( 1 + (0.889 + 0.456i)T \)
67 \( 1 + (-0.717 + 0.696i)T \)
71 \( 1 + (-0.984 + 0.176i)T \)
73 \( 1 + (0.147 + 0.989i)T \)
79 \( 1 + (-0.533 + 0.845i)T \)
83 \( 1 + (0.582 + 0.812i)T \)
89 \( 1 + (-0.0887 + 0.996i)T \)
97 \( 1 + (0.972 - 0.234i)T \)
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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

−29.56089160139314936649387398743, −28.6615796697387668102145934976, −27.780033853283801791295845441239, −26.99181753337045554267924341836, −25.75205584448599709362845025822, −23.77577972751613793207475583865, −23.39374520195714693205208598478, −22.16933337396029790480149057260, −21.47408343410330170402742772899, −20.626545456782404856898972088065, −18.8050178184822034029161472936, −18.32075722697090029854965284298, −17.446279050525382625414167140887, −15.66553624934752494590426145309, −14.66412060572372816489441199024, −13.39793614040457656235830772416, −12.06781846190246358974005323816, −11.16352699179653475160591641396, −10.57233941663903379873694385124, −9.25178882837053935725686523062, −7.39649706211407629897315505280, −5.765803641418587985563571700068, −4.84659526816327171162608610328, −3.28887046010257391643689082, −1.674596444549819028620227850462, 0.99893311197333833821767737666, 4.03396391837051352960027608689, 5.24397231462147125923807393074, 5.84436986474611021394878632105, 7.6068837364758121128736834996, 8.28840686622590739105232955764, 10.0233782849603465938965792926, 11.43525438176510646009279344636, 12.74947171228173669674499647359, 13.420956511993089820095198432822, 14.90776276680272407927410601918, 16.316151608551605687160708614098, 16.66702873723200095919694368487, 17.85968057784257362664861515703, 18.60391325497190687945282224393, 20.74725250092798723502453408262, 21.33977418288544246574646365480, 22.92199905521736743063887719563, 23.50326672167994242775978970643, 24.35143929048912225227108695251, 25.13364774585220059487984248562, 26.661377495704739991489138847542, 27.59853996459979756603621993608, 28.340748702950931578376796342220, 29.66599547055154516440230610338

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