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Properties

Label 1-107-107.41-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.564 + 0.825i$
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  + (−0.717 + 0.696i)2-s + (−0.861 + 0.508i)3-s + (0.0296 − 0.999i)4-s + (−0.0887 − 0.996i)5-s + (0.263 − 0.964i)6-s + (0.147 + 0.989i)7-s + (0.674 + 0.737i)8-s + (0.482 − 0.875i)9-s + (0.757 + 0.652i)10-s + (−0.320 + 0.947i)11-s + (0.482 + 0.875i)12-s + (−0.205 + 0.978i)13-s + (−0.794 − 0.606i)14-s + (0.582 + 0.812i)15-s + (−0.998 − 0.0592i)16-s + (0.937 − 0.348i)17-s + ⋯
L(s)  = 1  + (−0.717 + 0.696i)2-s + (−0.861 + 0.508i)3-s + (0.0296 − 0.999i)4-s + (−0.0887 − 0.996i)5-s + (0.263 − 0.964i)6-s + (0.147 + 0.989i)7-s + (0.674 + 0.737i)8-s + (0.482 − 0.875i)9-s + (0.757 + 0.652i)10-s + (−0.320 + 0.947i)11-s + (0.482 + 0.875i)12-s + (−0.205 + 0.978i)13-s + (−0.794 − 0.606i)14-s + (0.582 + 0.812i)15-s + (−0.998 − 0.0592i)16-s + (0.937 − 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2090606029 + 0.3959750164i\)
\(L(\frac12)\) \(\approx\) \(0.2090606029 + 0.3959750164i\)
\(L(1)\) \(\approx\) \(0.4612597140 + 0.2827412148i\)
\(L(1)\) \(\approx\) \(0.4612597140 + 0.2827412148i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.717 + 0.696i)T \)
3 \( 1 + (-0.861 + 0.508i)T \)
5 \( 1 + (-0.0887 - 0.996i)T \)
7 \( 1 + (0.147 + 0.989i)T \)
11 \( 1 + (-0.320 + 0.947i)T \)
13 \( 1 + (-0.205 + 0.978i)T \)
17 \( 1 + (0.937 - 0.348i)T \)
19 \( 1 + (-0.915 + 0.403i)T \)
23 \( 1 + (-0.794 + 0.606i)T \)
29 \( 1 + (0.889 + 0.456i)T \)
31 \( 1 + (0.375 + 0.926i)T \)
37 \( 1 + (-0.533 + 0.845i)T \)
41 \( 1 + (0.829 + 0.558i)T \)
43 \( 1 + (-0.0887 + 0.996i)T \)
47 \( 1 + (0.829 - 0.558i)T \)
53 \( 1 + (-0.717 - 0.696i)T \)
59 \( 1 + (0.889 - 0.456i)T \)
61 \( 1 + (0.147 - 0.989i)T \)
67 \( 1 + (0.674 - 0.737i)T \)
71 \( 1 + (-0.861 - 0.508i)T \)
73 \( 1 + (-0.430 + 0.902i)T \)
79 \( 1 + (0.992 - 0.118i)T \)
83 \( 1 + (-0.956 - 0.292i)T \)
89 \( 1 + (0.263 + 0.964i)T \)
97 \( 1 + (0.757 + 0.652i)T \)
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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

−29.70770707777695689915080050757, −28.25246563633829687737012452728, −27.300020244360054297119850066525, −26.518879404558117687627460300788, −25.42169302398971671794700357527, −23.96070609777458236973304633179, −22.91493645426032074460148815266, −22.05644534776815055693002824630, −20.95956518816821219136933337214, −19.486649429514920320163418192945, −18.84474425360696791535659364673, −17.75031628473686533403416132344, −17.07200617874074369257004594817, −15.8814423742837152183131826767, −14.03007550838238020420434187505, −12.930179765481243187788618935857, −11.73233416776612953975759037959, −10.50704053989524855496698058640, −10.43577763858134777392067072191, −8.116154842182669851055935896456, −7.32324335903733705194177375732, −6.02786613102082168570620949367, −4.01407504792652699415505521723, −2.52045104360387734604970637270, −0.64671241067221484675477653182, 1.640263792732000449955340582194, 4.578977707885724114465047685119, 5.35720871838217397793145947476, 6.55420557417400550251855578388, 8.11758360194280173198849882806, 9.29189929747154889971398560381, 10.07605633214558025134156129235, 11.67040748793130792534480154702, 12.48879645298469888152409260616, 14.44531288490802364457940412535, 15.61017670644686157018025707940, 16.26925093033974865965690078798, 17.2640763781273161959200448424, 18.12568416772901808898626663942, 19.30167820768987865730346346609, 20.71518395224897543926067263675, 21.63007232337835576963965915002, 23.17204978507607580323595000901, 23.780124649162613130746871921, 24.92479356698924412955607825156, 25.81565317521320392541103898532, 27.2130553456421328727293255877, 27.97987198525804354762706300249, 28.46539271535674378501956164999, 29.40426931104827574980608275932

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