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Properties

Label 1-643-643.8-r1-0-0
Degree $1$
Conductor $643$
Sign $0.996 + 0.0855i$
Analytic cond. $69.0999$
Root an. cond. $69.0999$
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  + (0.877 − 0.478i)2-s + (0.815 + 0.578i)3-s + (0.541 − 0.840i)4-s + (0.566 + 0.824i)5-s + (0.993 + 0.117i)6-s + (0.863 − 0.504i)7-s + (0.0733 − 0.997i)8-s + (0.331 + 0.943i)9-s + (0.891 + 0.452i)10-s + (0.742 + 0.669i)11-s + (0.928 − 0.372i)12-s + (−0.386 − 0.922i)13-s + (0.516 − 0.856i)14-s + (−0.0146 + 0.999i)15-s + (−0.412 − 0.910i)16-s + (0.246 + 0.969i)17-s + ⋯
L(s)  = 1  + (0.877 − 0.478i)2-s + (0.815 + 0.578i)3-s + (0.541 − 0.840i)4-s + (0.566 + 0.824i)5-s + (0.993 + 0.117i)6-s + (0.863 − 0.504i)7-s + (0.0733 − 0.997i)8-s + (0.331 + 0.943i)9-s + (0.891 + 0.452i)10-s + (0.742 + 0.669i)11-s + (0.928 − 0.372i)12-s + (−0.386 − 0.922i)13-s + (0.516 − 0.856i)14-s + (−0.0146 + 0.999i)15-s + (−0.412 − 0.910i)16-s + (0.246 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.996 + 0.0855i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ 0.996 + 0.0855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.806981175 + 0.2917951371i\)
\(L(\frac12)\) \(\approx\) \(6.806981175 + 0.2917951371i\)
\(L(1)\) \(\approx\) \(2.962687177 - 0.06272661063i\)
\(L(1)\) \(\approx\) \(2.962687177 - 0.06272661063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (0.877 - 0.478i)T \)
3 \( 1 + (0.815 + 0.578i)T \)
5 \( 1 + (0.566 + 0.824i)T \)
7 \( 1 + (0.863 - 0.504i)T \)
11 \( 1 + (0.742 + 0.669i)T \)
13 \( 1 + (-0.386 - 0.922i)T \)
17 \( 1 + (0.246 + 0.969i)T \)
19 \( 1 + (0.978 - 0.204i)T \)
23 \( 1 + (0.0440 + 0.999i)T \)
29 \( 1 + (-0.742 - 0.669i)T \)
31 \( 1 + (-0.0146 - 0.999i)T \)
37 \( 1 + (0.701 + 0.712i)T \)
41 \( 1 + (0.999 + 0.0293i)T \)
43 \( 1 + (-0.491 - 0.870i)T \)
47 \( 1 + (-0.863 + 0.504i)T \)
53 \( 1 + (-0.989 + 0.146i)T \)
59 \( 1 + (0.815 - 0.578i)T \)
61 \( 1 + (-0.636 - 0.771i)T \)
67 \( 1 + (0.189 + 0.981i)T \)
71 \( 1 + (0.613 + 0.789i)T \)
73 \( 1 + (-0.636 - 0.771i)T \)
79 \( 1 + (0.613 - 0.789i)T \)
83 \( 1 + (-0.0146 + 0.999i)T \)
89 \( 1 + (-0.877 - 0.478i)T \)
97 \( 1 + (0.722 + 0.691i)T \)
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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

−22.71791208554415542480925846381, −21.63011825364806903075503791912, −21.17995710751083919050698675408, −20.40779210176949793073372871951, −19.675606736228867914502696629291, −18.39238275153854028150741440429, −17.76730580673772045861052853703, −16.59557905583788152849195740840, −16.13417886142163553240508805937, −14.73496781506614950207180177256, −14.25541842162123233641236203082, −13.75362659037303723231626465385, −12.6932141562602901930643390756, −12.03985195901027377447082389928, −11.313114422543558060511556557543, −9.43959769261278478285802002003, −8.8511148130303891784982643364, −8.027119588515780058366920720545, −7.08816518531212379159108905637, −6.13285539974343596107928511454, −5.160747228320058933323268029965, −4.32636801700214958667941825723, −3.10546994551993621464890327991, −2.09530896881327854799191615144, −1.188497497818872943851856734263, 1.39699010439422442551006901156, 2.20550585143435349555283442151, 3.26672353532817553256367183907, 4.00523568227478239637747404112, 5.00771608056303511272975422033, 5.92004423338400919291827610763, 7.22988466808845165768446263882, 7.8832005324378943979386281279, 9.59506624387725652702638478101, 9.90428006047915638443881541113, 10.909216504890800010463497262446, 11.5481982765959291014838631708, 12.91582312325989350523219338624, 13.64883227880840573713590620215, 14.426106260892257198191223273644, 14.92686287165916104907269266989, 15.48919943634517659723654767348, 16.95045073121077470804425894845, 17.76296540011245049767634442169, 18.90197819957669742429407628867, 19.72645194843800813107522214895, 20.43019640413013783711009927466, 21.013548591455023898270801886989, 22.01967751818758108283824135659, 22.29415511077629967614112920238

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