Properties

Label 1-107-107.16-r0-0-0
Degree $1$
Conductor $107$
Sign $0.550 - 0.834i$
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  + (0.972 + 0.234i)2-s + (−0.630 − 0.776i)3-s + (0.889 + 0.456i)4-s + (0.147 − 0.989i)5-s + (−0.430 − 0.902i)6-s + (−0.717 − 0.696i)7-s + (0.757 + 0.652i)8-s + (−0.205 + 0.978i)9-s + (0.375 − 0.926i)10-s + (0.482 − 0.875i)11-s + (−0.205 − 0.978i)12-s + (−0.984 − 0.176i)13-s + (−0.533 − 0.845i)14-s + (−0.861 + 0.508i)15-s + (0.582 + 0.812i)16-s + (0.829 + 0.558i)17-s + ⋯
L(s)  = 1  + (0.972 + 0.234i)2-s + (−0.630 − 0.776i)3-s + (0.889 + 0.456i)4-s + (0.147 − 0.989i)5-s + (−0.430 − 0.902i)6-s + (−0.717 − 0.696i)7-s + (0.757 + 0.652i)8-s + (−0.205 + 0.978i)9-s + (0.375 − 0.926i)10-s + (0.482 − 0.875i)11-s + (−0.205 − 0.978i)12-s + (−0.984 − 0.176i)13-s + (−0.533 − 0.845i)14-s + (−0.861 + 0.508i)15-s + (0.582 + 0.812i)16-s + (0.829 + 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.550 - 0.834i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269919398 - 0.6833834761i\)
\(L(\frac12)\) \(\approx\) \(1.269919398 - 0.6833834761i\)
\(L(1)\) \(\approx\) \(1.372173259 - 0.4026527371i\)
\(L(1)\) \(\approx\) \(1.372173259 - 0.4026527371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.972 + 0.234i)T \)
3 \( 1 + (-0.630 - 0.776i)T \)
5 \( 1 + (0.147 - 0.989i)T \)
7 \( 1 + (-0.717 - 0.696i)T \)
11 \( 1 + (0.482 - 0.875i)T \)
13 \( 1 + (-0.984 - 0.176i)T \)
17 \( 1 + (0.829 + 0.558i)T \)
19 \( 1 + (0.937 - 0.348i)T \)
23 \( 1 + (-0.533 + 0.845i)T \)
29 \( 1 + (0.263 + 0.964i)T \)
31 \( 1 + (0.992 + 0.118i)T \)
37 \( 1 + (-0.915 + 0.403i)T \)
41 \( 1 + (-0.998 - 0.0592i)T \)
43 \( 1 + (0.147 + 0.989i)T \)
47 \( 1 + (-0.998 + 0.0592i)T \)
53 \( 1 + (0.972 - 0.234i)T \)
59 \( 1 + (0.263 - 0.964i)T \)
61 \( 1 + (-0.717 + 0.696i)T \)
67 \( 1 + (0.757 - 0.652i)T \)
71 \( 1 + (-0.630 + 0.776i)T \)
73 \( 1 + (0.674 + 0.737i)T \)
79 \( 1 + (-0.320 - 0.947i)T \)
83 \( 1 + (0.0296 - 0.999i)T \)
89 \( 1 + (-0.430 + 0.902i)T \)
97 \( 1 + (0.375 - 0.926i)T \)
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

−29.73158208610408972666148822382, −28.91534656128882999963784391555, −28.02380926399377372300508766648, −26.71472138632281259795456986662, −25.61184264650150091731028829882, −24.56728030903674450937016529490, −22.87676856396647595228294337281, −22.64463027795529313541196819232, −21.83261034201776211645048884207, −20.798629047741705386442045638196, −19.53140645335653143835132866545, −18.34290610182196342343392533128, −16.893422537004154565254853899316, −15.71085673915918596358479295041, −14.939892080352050751241112215079, −14.01246544741679938420941377715, −12.156653614823380956393951592932, −11.8378845971726850611760314299, −10.16876540891555965296171223241, −9.75770472087814338846844006556, −7.1116551457051048032233626007, −6.15437246407166294200911667877, −5.03810264735152694158424681400, −3.64626824611710604273522324078, −2.48676698028241642966313457950, 1.32213256533463748172892092998, 3.301702489607463011267843698301, 4.90379931371295222406076779234, 5.8757227598377724904195086228, 7.02007195813054651171865672831, 8.15067205969513233337293836687, 10.07181992041455226067012414418, 11.62648790652100496015349298438, 12.40932262381245798906413611478, 13.35713523856151671142631020749, 14.10659954932442454889936437919, 15.96987513083359673800601826937, 16.72269796905851263412475929575, 17.434090596019456094502180606949, 19.39312235123085965565776337154, 19.99444815559015627290519337136, 21.48981835837105771117623399244, 22.37431700790442009703188477775, 23.4365855540433404304893622108, 24.211392702368876122354425005802, 24.87411379201748863207497395673, 26.03762685819902458015860692641, 27.63240893245221191451800303950, 28.99912780523756275420070349475, 29.44318838112679249390113010848

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