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Properties

Label 1-643-643.322-r1-0-0
Degree $1$
Conductor $643$
Sign $0.997 + 0.0737i$
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  + (−0.636 + 0.771i)2-s + (0.978 − 0.204i)3-s + (−0.189 − 0.981i)4-s + (0.948 − 0.317i)5-s + (−0.465 + 0.884i)6-s + (0.984 + 0.175i)7-s + (0.877 + 0.478i)8-s + (0.916 − 0.399i)9-s + (−0.358 + 0.933i)10-s + (−0.275 + 0.961i)11-s + (−0.386 − 0.922i)12-s + (0.131 − 0.991i)13-s + (−0.761 + 0.647i)14-s + (0.863 − 0.504i)15-s + (−0.928 + 0.372i)16-s + (0.904 − 0.426i)17-s + ⋯
L(s)  = 1  + (−0.636 + 0.771i)2-s + (0.978 − 0.204i)3-s + (−0.189 − 0.981i)4-s + (0.948 − 0.317i)5-s + (−0.465 + 0.884i)6-s + (0.984 + 0.175i)7-s + (0.877 + 0.478i)8-s + (0.916 − 0.399i)9-s + (−0.358 + 0.933i)10-s + (−0.275 + 0.961i)11-s + (−0.386 − 0.922i)12-s + (0.131 − 0.991i)13-s + (−0.761 + 0.647i)14-s + (0.863 − 0.504i)15-s + (−0.928 + 0.372i)16-s + (0.904 − 0.426i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.997 + 0.0737i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ 0.997 + 0.0737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.298519431 + 0.1217535830i\)
\(L(\frac12)\) \(\approx\) \(3.298519431 + 0.1217535830i\)
\(L(1)\) \(\approx\) \(1.540707955 + 0.1938433537i\)
\(L(1)\) \(\approx\) \(1.540707955 + 0.1938433537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.636 + 0.771i)T \)
3 \( 1 + (0.978 - 0.204i)T \)
5 \( 1 + (0.948 - 0.317i)T \)
7 \( 1 + (0.984 + 0.175i)T \)
11 \( 1 + (-0.275 + 0.961i)T \)
13 \( 1 + (0.131 - 0.991i)T \)
17 \( 1 + (0.904 - 0.426i)T \)
19 \( 1 + (-0.439 - 0.898i)T \)
23 \( 1 + (-0.0146 + 0.999i)T \)
29 \( 1 + (0.275 - 0.961i)T \)
31 \( 1 + (0.863 + 0.504i)T \)
37 \( 1 + (0.965 - 0.261i)T \)
41 \( 1 + (-0.491 + 0.870i)T \)
43 \( 1 + (-0.938 + 0.345i)T \)
47 \( 1 + (-0.984 - 0.175i)T \)
53 \( 1 + (0.541 - 0.840i)T \)
59 \( 1 + (0.978 + 0.204i)T \)
61 \( 1 + (-0.957 + 0.289i)T \)
67 \( 1 + (-0.832 - 0.554i)T \)
71 \( 1 + (-0.218 + 0.975i)T \)
73 \( 1 + (-0.957 + 0.289i)T \)
79 \( 1 + (-0.218 - 0.975i)T \)
83 \( 1 + (0.863 - 0.504i)T \)
89 \( 1 + (0.636 + 0.771i)T \)
97 \( 1 + (-0.701 - 0.712i)T \)
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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

−22.1417993760302175252488629601, −21.42858475181834429150825103351, −21.03222497612249220290193568897, −20.45266937142691067601617610620, −19.23311232019632588934087720835, −18.66296325867192006471366164067, −18.11150124117498582438459718082, −16.815965695227379452354050643724, −16.48502419278224763903695564961, −14.90546559813701783525609746724, −14.13083598026273548617166472059, −13.64041359907075945229234763953, −12.62101884776769884201500715673, −11.50697293496855424409483868512, −10.47306296768302043428241280949, −10.118742756213689201855254339745, −8.912703316076978779871742606844, −8.42275890117818533951142524076, −7.55187875772665539443352455157, −6.34059763416136745542558134059, −4.87504030443867294744994210399, −3.8467346488439734209120020112, −2.86933621699144580155120726649, −1.93048012134960747028644803268, −1.216761271005851941682048818139, 0.96153050416775536181573575206, 1.80018515405559440878939944050, 2.76026598766959133778702478223, 4.59301441999888420461312942601, 5.237405085468146814758966896178, 6.37009928162244089234222928517, 7.478448810288684161699818849051, 8.07608965720628930912053452671, 8.88176622824570538113055875371, 9.830943015787861000597512119039, 10.24956869392193602582367164194, 11.713965138950387307587236116612, 13.09770327486794371640366585346, 13.57318220512968221788443674242, 14.6669377301235571641375625981, 15.022957923966850670518434253375, 15.94623786411381215152872605065, 17.17203530032869119166509186315, 17.91918414574069153731243114402, 18.16276834676240458639751951159, 19.401343831556031158173682194709, 20.17961044191553092082120370820, 20.90250682275520130949166291467, 21.612644091397179093243941147791, 23.04316722399686168783424321882

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