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Properties

Label 1-107-107.27-r0-0-0
Degree $1$
Conductor $107$
Sign $0.858 - 0.513i$
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  + (0.992 − 0.118i)2-s + (−0.430 − 0.902i)3-s + (0.972 − 0.234i)4-s + (0.757 + 0.652i)5-s + (−0.533 − 0.845i)6-s + (0.375 + 0.926i)7-s + (0.937 − 0.348i)8-s + (−0.630 + 0.776i)9-s + (0.829 + 0.558i)10-s + (−0.861 − 0.508i)11-s + (−0.630 − 0.776i)12-s + (−0.0887 − 0.996i)13-s + (0.482 + 0.875i)14-s + (0.263 − 0.964i)15-s + (0.889 − 0.456i)16-s + (−0.956 + 0.292i)17-s + ⋯
L(s)  = 1  + (0.992 − 0.118i)2-s + (−0.430 − 0.902i)3-s + (0.972 − 0.234i)4-s + (0.757 + 0.652i)5-s + (−0.533 − 0.845i)6-s + (0.375 + 0.926i)7-s + (0.937 − 0.348i)8-s + (−0.630 + 0.776i)9-s + (0.829 + 0.558i)10-s + (−0.861 − 0.508i)11-s + (−0.630 − 0.776i)12-s + (−0.0887 − 0.996i)13-s + (0.482 + 0.875i)14-s + (0.263 − 0.964i)15-s + (0.889 − 0.456i)16-s + (−0.956 + 0.292i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.858 - 0.513i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.858 - 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.667174661 - 0.4602857509i\)
\(L(\frac12)\) \(\approx\) \(1.667174661 - 0.4602857509i\)
\(L(1)\) \(\approx\) \(1.636236814 - 0.3365693556i\)
\(L(1)\) \(\approx\) \(1.636236814 - 0.3365693556i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.992 - 0.118i)T \)
3 \( 1 + (-0.430 - 0.902i)T \)
5 \( 1 + (0.757 + 0.652i)T \)
7 \( 1 + (0.375 + 0.926i)T \)
11 \( 1 + (-0.861 - 0.508i)T \)
13 \( 1 + (-0.0887 - 0.996i)T \)
17 \( 1 + (-0.956 + 0.292i)T \)
19 \( 1 + (-0.984 - 0.176i)T \)
23 \( 1 + (0.482 - 0.875i)T \)
29 \( 1 + (-0.794 + 0.606i)T \)
31 \( 1 + (-0.998 + 0.0592i)T \)
37 \( 1 + (-0.205 + 0.978i)T \)
41 \( 1 + (0.0296 + 0.999i)T \)
43 \( 1 + (0.757 - 0.652i)T \)
47 \( 1 + (0.0296 - 0.999i)T \)
53 \( 1 + (0.992 + 0.118i)T \)
59 \( 1 + (-0.794 - 0.606i)T \)
61 \( 1 + (0.375 - 0.926i)T \)
67 \( 1 + (0.937 + 0.348i)T \)
71 \( 1 + (-0.430 + 0.902i)T \)
73 \( 1 + (-0.915 + 0.403i)T \)
79 \( 1 + (0.582 + 0.812i)T \)
83 \( 1 + (-0.717 - 0.696i)T \)
89 \( 1 + (-0.533 + 0.845i)T \)
97 \( 1 + (0.829 + 0.558i)T \)
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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

−29.45839426846926546531070230588, −28.98914034722261047780766271272, −27.85750861550669848897459852412, −26.4084781441761692611866975709, −25.67666637469308287441807177980, −24.20345871115954168547200317751, −23.50825812222270106352944834406, −22.46475569707937080419464997073, −21.18166250877657452774896583789, −20.97368324793647411949876156277, −19.85270271291877284006009550371, −17.627484806295837600987822584974, −16.86218901904840584364036618300, −15.96240639596716123016037944362, −14.788836744913086653555044342612, −13.710523227527900237566779700806, −12.73092003804561654009828805299, −11.302118359903575495910758353493, −10.4416416191623721759427679399, −9.11105016267148917439430353322, −7.287824595989592465300535534548, −5.89412011675158836427635535248, −4.79193415998565058953109822665, −4.064027364202516211731491361088, −2.07757707191688362520819902794, 2.01500177086074828935697629938, 2.83904418420449828991444945597, 5.16353906842281212345213323793, 5.914712868392308904490730507405, 6.95250532720122964139619308146, 8.39595963719871242454233302318, 10.58408784422307144098398359943, 11.21717829315659404233552840359, 12.72045066218388865637963117229, 13.19911172194833630238358678729, 14.50818345149917980268623582946, 15.40516947685306453801233326284, 16.974471548004056968984047762807, 18.17581584151067922035675350566, 18.91494455135061723872630758938, 20.34926843613458124632085043046, 21.66174502933703204058604209324, 22.21962876441275541269564886208, 23.32262343904686560732300217532, 24.367023749010195061727687137352, 25.07307105661154782214919232908, 26.0174928631109492678096588965, 27.9272513538445308585093107039, 28.988529942671927209034500425260, 29.56915212610216589517484898141

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