Properties

Label 1-107-107.29-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.545 + 0.837i$
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  + (−0.320 + 0.947i)2-s + (0.674 + 0.737i)3-s + (−0.794 − 0.606i)4-s + (0.375 + 0.926i)5-s + (−0.915 + 0.403i)6-s + (0.992 − 0.118i)7-s + (0.829 − 0.558i)8-s + (−0.0887 + 0.996i)9-s + (−0.998 + 0.0592i)10-s + (−0.630 − 0.776i)11-s + (−0.0887 − 0.996i)12-s + (0.147 + 0.989i)13-s + (−0.205 + 0.978i)14-s + (−0.430 + 0.902i)15-s + (0.263 + 0.964i)16-s + (0.0296 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.320 + 0.947i)2-s + (0.674 + 0.737i)3-s + (−0.794 − 0.606i)4-s + (0.375 + 0.926i)5-s + (−0.915 + 0.403i)6-s + (0.992 − 0.118i)7-s + (0.829 − 0.558i)8-s + (−0.0887 + 0.996i)9-s + (−0.998 + 0.0592i)10-s + (−0.630 − 0.776i)11-s + (−0.0887 − 0.996i)12-s + (0.147 + 0.989i)13-s + (−0.205 + 0.978i)14-s + (−0.430 + 0.902i)15-s + (0.263 + 0.964i)16-s + (0.0296 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.545 + 0.837i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.545 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5257477516 + 0.9700843929i\)
\(L(\frac12)\) \(\approx\) \(0.5257477516 + 0.9700843929i\)
\(L(1)\) \(\approx\) \(0.8040394657 + 0.7560004586i\)
\(L(1)\) \(\approx\) \(0.8040394657 + 0.7560004586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (-0.320 + 0.947i)T \)
3 \( 1 + (0.674 + 0.737i)T \)
5 \( 1 + (0.375 + 0.926i)T \)
7 \( 1 + (0.992 - 0.118i)T \)
11 \( 1 + (-0.630 - 0.776i)T \)
13 \( 1 + (0.147 + 0.989i)T \)
17 \( 1 + (0.0296 - 0.999i)T \)
19 \( 1 + (-0.956 - 0.292i)T \)
23 \( 1 + (-0.205 - 0.978i)T \)
29 \( 1 + (-0.533 - 0.845i)T \)
31 \( 1 + (0.582 + 0.812i)T \)
37 \( 1 + (-0.984 + 0.176i)T \)
41 \( 1 + (0.889 + 0.456i)T \)
43 \( 1 + (0.375 - 0.926i)T \)
47 \( 1 + (0.889 - 0.456i)T \)
53 \( 1 + (-0.320 - 0.947i)T \)
59 \( 1 + (-0.533 + 0.845i)T \)
61 \( 1 + (0.992 + 0.118i)T \)
67 \( 1 + (0.829 + 0.558i)T \)
71 \( 1 + (0.674 - 0.737i)T \)
73 \( 1 + (0.937 + 0.348i)T \)
79 \( 1 + (-0.861 - 0.508i)T \)
83 \( 1 + (0.972 + 0.234i)T \)
89 \( 1 + (-0.915 - 0.403i)T \)
97 \( 1 + (-0.998 + 0.0592i)T \)
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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

−29.48021746988522000313779159628, −28.15748775221285271930264430728, −27.65810413867372337244980793019, −26.09597664186657213402420844809, −25.34122882439391574492383092722, −24.15340878447569015861395783662, −23.21204002511532681774248864790, −21.48488109250140618609414259404, −20.71383232146060077413679563473, −20.08967167483498541416052634559, −18.92519429776716998201275325944, −17.713542239505452621605499123396, −17.3431871527459588120858846246, −15.2648378249132869020310237595, −13.98126484550951787150035590079, −12.82358994300882385398881352312, −12.41354723129172430503307212953, −10.831988066192814252130311898041, −9.52245840180590202976449572619, −8.35617553744417313346182043124, −7.7835080842956350796158527851, −5.491199178644841171755047342312, −4.061720330492741533795995078043, −2.30078684446266344072202094276, −1.37735981651585476007870269764, 2.32288849252184122943655505922, 4.13470349325384689445114146723, 5.34234988896683282635149314884, 6.82983359490346506829501536390, 8.062371214412944620215614385519, 9.00820109514020164050422545178, 10.2884644818284913827293249184, 11.1175601475517495887522056714, 13.68786787335919558092789101965, 14.15716083420301047381055985074, 15.09984248178458319095789804138, 16.11006360000392664548428211305, 17.23981911302371827026730225312, 18.47117807726241158064400300465, 19.17348927184739990493177248833, 20.86139450617605822773312952941, 21.663708590567451983819659970, 22.80752648556988450626645722875, 24.04613082831790073788368358085, 25.01727110726965690494568970920, 26.147087894200418497530050641203, 26.633404762275850355469690068121, 27.43908882134198998977789010362, 28.656268209483219455314033425809, 30.22106756305205626844085294057

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