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Properties

Label 1-643-643.19-r1-0-0
Degree $1$
Conductor $643$
Sign $0.243 + 0.969i$
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  + (−0.439 + 0.898i)2-s + (−0.891 + 0.452i)3-s + (−0.613 − 0.789i)4-s + (−0.491 − 0.870i)5-s + (−0.0146 − 0.999i)6-s + (0.980 − 0.194i)7-s + (0.978 − 0.204i)8-s + (0.590 − 0.807i)9-s + (0.998 − 0.0586i)10-s + (0.533 − 0.845i)11-s + (0.904 + 0.426i)12-s + (−0.367 + 0.929i)13-s + (−0.256 + 0.966i)14-s + (0.832 + 0.554i)15-s + (−0.246 + 0.969i)16-s + (−0.882 − 0.470i)17-s + ⋯
L(s)  = 1  + (−0.439 + 0.898i)2-s + (−0.891 + 0.452i)3-s + (−0.613 − 0.789i)4-s + (−0.491 − 0.870i)5-s + (−0.0146 − 0.999i)6-s + (0.980 − 0.194i)7-s + (0.978 − 0.204i)8-s + (0.590 − 0.807i)9-s + (0.998 − 0.0586i)10-s + (0.533 − 0.845i)11-s + (0.904 + 0.426i)12-s + (−0.367 + 0.929i)13-s + (−0.256 + 0.966i)14-s + (0.832 + 0.554i)15-s + (−0.246 + 0.969i)16-s + (−0.882 − 0.470i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.243 + 0.969i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ 0.243 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5928324909 + 0.4625479455i\)
\(L(\frac12)\) \(\approx\) \(0.5928324909 + 0.4625479455i\)
\(L(1)\) \(\approx\) \(0.5610966088 + 0.1872006377i\)
\(L(1)\) \(\approx\) \(0.5610966088 + 0.1872006377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.439 + 0.898i)T \)
3 \( 1 + (-0.891 + 0.452i)T \)
5 \( 1 + (-0.491 - 0.870i)T \)
7 \( 1 + (0.980 - 0.194i)T \)
11 \( 1 + (0.533 - 0.845i)T \)
13 \( 1 + (-0.367 + 0.929i)T \)
17 \( 1 + (-0.882 - 0.470i)T \)
19 \( 1 + (-0.987 - 0.155i)T \)
23 \( 1 + (-0.755 + 0.655i)T \)
29 \( 1 + (0.999 + 0.0391i)T \)
31 \( 1 + (-0.896 - 0.443i)T \)
37 \( 1 + (0.227 + 0.973i)T \)
41 \( 1 + (0.991 + 0.126i)T \)
43 \( 1 + (0.131 + 0.991i)T \)
47 \( 1 + (0.322 - 0.946i)T \)
53 \( 1 + (-0.804 + 0.594i)T \)
59 \( 1 + (0.837 - 0.545i)T \)
61 \( 1 + (0.151 - 0.988i)T \)
67 \( 1 + (-0.218 + 0.975i)T \)
71 \( 1 + (-0.275 + 0.961i)T \)
73 \( 1 + (-0.931 + 0.363i)T \)
79 \( 1 + (-0.694 + 0.719i)T \)
83 \( 1 + (-0.896 + 0.443i)T \)
89 \( 1 + (0.558 - 0.829i)T \)
97 \( 1 + (-0.986 - 0.165i)T \)
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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

−22.319664696251474535919749715345, −21.93455390474921858548899718505, −20.867862349070427622355174424, −19.79094378358515726041184923562, −19.29638935113731585646182620052, −18.17464275694817830356931230651, −17.792278502064372948647648204661, −17.2789390860818772278111965587, −16.027512900894493421909551102060, −14.9238136702909666556502402109, −14.13587552678123184000570652703, −12.779970467946556863157636685396, −12.27595078057308238565868875235, −11.46960926495087479811504696953, −10.65123739866548553352721951753, −10.31908876060570799666030245123, −8.79312319231176844468645537561, −7.833689475811365667477460277279, −7.19007233846794363083285416541, −6.05405041763899702953605974872, −4.676062366947886608275552604573, −4.06859800785021322754381810136, −2.465072717630965133891701776473, −1.80307354788340385864246335371, −0.40346227454344483038737000570, 0.64022159180501325957447075328, 1.63649537654873020069781041964, 4.17182998009446452367666953885, 4.464215740968699516400624120688, 5.435894328138935709547516674902, 6.364820014140082830056472485319, 7.299218810289945637867811243642, 8.398567185182775695124805845338, 9.01785162983901811154322937478, 9.94972287160190860266162793221, 11.271506566474729548521502609743, 11.48014068492560590142484183716, 12.80980004473608579287061414264, 13.90515730297864473925368289667, 14.7475903187682744292077667764, 15.71749711671563657421545667242, 16.29704282402444229759904557837, 17.07563242295377122323324355776, 17.50538799302845150510107748411, 18.51576343539571788392584447157, 19.46781515833722631231114765773, 20.330419632632055089243384574342, 21.49199052728149650666481860022, 22.00544785313850731662729699605, 23.275337831358680224793304506971

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