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Properties

Label 1-644-644.395-r0-0-0
Degree $1$
Conductor $644$
Sign $0.961 - 0.275i$
Analytic cond. $2.99072$
Root an. cond. $2.99072$
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  + (−0.888 − 0.458i)3-s + (0.995 + 0.0950i)5-s + (0.580 + 0.814i)9-s + (0.327 − 0.945i)11-s + (0.959 + 0.281i)13-s + (−0.841 − 0.540i)15-s + (−0.928 + 0.371i)17-s + (0.928 + 0.371i)19-s + (0.981 + 0.189i)25-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (0.0475 − 0.998i)31-s + (−0.723 + 0.690i)33-s + (0.580 + 0.814i)37-s + (−0.723 − 0.690i)39-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)3-s + (0.995 + 0.0950i)5-s + (0.580 + 0.814i)9-s + (0.327 − 0.945i)11-s + (0.959 + 0.281i)13-s + (−0.841 − 0.540i)15-s + (−0.928 + 0.371i)17-s + (0.928 + 0.371i)19-s + (0.981 + 0.189i)25-s + (−0.142 − 0.989i)27-s + (−0.142 + 0.989i)29-s + (0.0475 − 0.998i)31-s + (−0.723 + 0.690i)33-s + (0.580 + 0.814i)37-s + (−0.723 − 0.690i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(2.99072\)
Root analytic conductor: \(2.99072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 644,\ (0:\ ),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344373805 - 0.1888403036i\)
\(L(\frac12)\) \(\approx\) \(1.344373805 - 0.1888403036i\)
\(L(1)\) \(\approx\) \(1.044231697 - 0.1124622886i\)
\(L(1)\) \(\approx\) \(1.044231697 - 0.1124622886i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.888 - 0.458i)T \)
5 \( 1 + (0.995 + 0.0950i)T \)
11 \( 1 + (0.327 - 0.945i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (-0.928 + 0.371i)T \)
19 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (0.0475 - 0.998i)T \)
37 \( 1 + (0.580 + 0.814i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.723 + 0.690i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (0.888 - 0.458i)T \)
67 \( 1 + (-0.981 - 0.189i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (0.786 - 0.618i)T \)
79 \( 1 + (-0.723 + 0.690i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−22.789983908465077082464749657231, −22.162034144025109539150871803924, −21.382870002434302557929284927784, −20.609656933466603197941863851538, −19.9067445992798112254318105146, −18.3145715160307208988043718962, −17.94002641077353074056730964849, −17.27420513398124473738615766204, −16.33864261318087451015275123335, −15.601340000329748878975313394098, −14.71275802119055752525847210559, −13.56130555917146432653473171423, −12.93024770753567776832521813581, −11.84184042766542828274567699317, −11.103858661124679102163190659491, −10.10000369258723615803564327612, −9.55697252034914379874826969625, −8.616725109094860353578214599565, −7.05032890461950875613315762077, −6.41264974240135256877244648781, −5.42735349345583877134648545609, −4.72913122610821096634347466620, −3.607789563830812501910910540786, −2.17077374861060987426339513760, −1.01443787913778898319121350317, 1.058694479128983936675455324214, 1.91670948392246050090814164231, 3.29519481882376794552684647453, 4.615363613109925019183996080308, 5.74442587461636412573497206288, 6.18442664110646819767666180027, 7.05151002705918585802984009240, 8.317442100323118673975985044032, 9.238914069666568090515631311092, 10.30960776861736368689680457875, 11.090528071395508978688210393063, 11.7458157053470931275204410085, 12.98609581200804332049420135303, 13.49639018441390870561177239655, 14.24390557878283659941887541145, 15.56566337000744940913356246546, 16.54392149598411573554965838989, 16.97895706482364090194725477564, 18.15099271422093550774267447664, 18.34284006148455809148755552847, 19.39224227744293249823523601343, 20.52315461765716629295341725058, 21.4332597724994808955851682630, 22.13077967594704710221047886082, 22.65404151324602405837429771995

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