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Properties

Label 1-643-643.2-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: $\chi_{643} (2, \cdot )$
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

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

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