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Properties

Label 1-643-643.418-r1-0-0
Degree $1$
Conductor $643$
Sign $-0.0812 + 0.996i$
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  + (−0.491 − 0.870i)2-s + (0.928 − 0.372i)3-s + (−0.516 + 0.856i)4-s + (0.0733 − 0.997i)5-s + (−0.780 − 0.625i)6-s + (0.102 − 0.994i)7-s + (0.999 + 0.0293i)8-s + (0.722 − 0.691i)9-s + (−0.904 + 0.426i)10-s + (−0.957 − 0.289i)11-s + (−0.160 + 0.986i)12-s + (−0.891 − 0.452i)13-s + (−0.916 + 0.399i)14-s + (−0.303 − 0.952i)15-s + (−0.465 − 0.884i)16-s + (−0.863 − 0.504i)17-s + ⋯
L(s)  = 1  + (−0.491 − 0.870i)2-s + (0.928 − 0.372i)3-s + (−0.516 + 0.856i)4-s + (0.0733 − 0.997i)5-s + (−0.780 − 0.625i)6-s + (0.102 − 0.994i)7-s + (0.999 + 0.0293i)8-s + (0.722 − 0.691i)9-s + (−0.904 + 0.426i)10-s + (−0.957 − 0.289i)11-s + (−0.160 + 0.986i)12-s + (−0.891 − 0.452i)13-s + (−0.916 + 0.399i)14-s + (−0.303 − 0.952i)15-s + (−0.465 − 0.884i)16-s + (−0.863 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0812 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0812 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.0812 + 0.996i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ -0.0812 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8164686636 - 0.8857743481i\)
\(L(\frac12)\) \(\approx\) \(-0.8164686636 - 0.8857743481i\)
\(L(1)\) \(\approx\) \(0.5206542684 - 0.7895287427i\)
\(L(1)\) \(\approx\) \(0.5206542684 - 0.7895287427i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.491 - 0.870i)T \)
3 \( 1 + (0.928 - 0.372i)T \)
5 \( 1 + (0.0733 - 0.997i)T \)
7 \( 1 + (0.102 - 0.994i)T \)
11 \( 1 + (-0.957 - 0.289i)T \)
13 \( 1 + (-0.891 - 0.452i)T \)
17 \( 1 + (-0.863 - 0.504i)T \)
19 \( 1 + (-0.386 - 0.922i)T \)
23 \( 1 + (0.798 - 0.601i)T \)
29 \( 1 + (0.957 + 0.289i)T \)
31 \( 1 + (-0.303 + 0.952i)T \)
37 \( 1 + (-0.590 + 0.807i)T \)
41 \( 1 + (0.815 - 0.578i)T \)
43 \( 1 + (0.978 - 0.204i)T \)
47 \( 1 + (-0.102 + 0.994i)T \)
53 \( 1 + (0.998 - 0.0586i)T \)
59 \( 1 + (0.928 + 0.372i)T \)
61 \( 1 + (-0.938 - 0.345i)T \)
67 \( 1 + (-0.761 + 0.647i)T \)
71 \( 1 + (0.965 - 0.261i)T \)
73 \( 1 + (-0.938 - 0.345i)T \)
79 \( 1 + (0.965 + 0.261i)T \)
83 \( 1 + (-0.303 - 0.952i)T \)
89 \( 1 + (0.491 - 0.870i)T \)
97 \( 1 + (-0.948 + 0.317i)T \)
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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

−23.3774773456088920299678183331, −22.48142716030811748050217996392, −21.66237978433739088585957270031, −20.985747632950909733275024069174, −19.50467677549545564394376207583, −19.22128560862117689987840434112, −18.33203649913790465934307623802, −17.70954614575718435549153206367, −16.512329885891790125176610026793, −15.46929527572573575163961656050, −15.14066506364414088059959055364, −14.49469408351083705050624974400, −13.623879184316690442283772727298, −12.62937900750062579818700019192, −11.12469466710087488088677868937, −10.26297477914133338927413035756, −9.56802513792332557040462750285, −8.71666498708648859752081621654, −7.83134478749589318750698988477, −7.15336940441470179046229715902, −6.0635587668179000972372656429, −5.07165332222872387631212794898, −4.01269821453453601006820464906, −2.555585306489844830110367278932, −2.00042753369500741222202392514, 0.3179790183140811415937606092, 1.0458526034310586836681183194, 2.31949379071544889633693638703, 3.05606173613448074744712904811, 4.37497351740975976429280340362, 4.924865332214113639560073301796, 6.96050440926774238183211880794, 7.6588866385506006907226072326, 8.56796938163170639263913871558, 9.11872326355961558457987569076, 10.164084880855296741207139639866, 10.8586078354438203842933790075, 12.1748939351636386362236405399, 12.91672995676363524577750485728, 13.40750201788914310879957104075, 14.18859731935850719332358804342, 15.55555772233255206101536959143, 16.4053168530779145812644201056, 17.51735148085312639981097828930, 17.859652839417930285117633332465, 19.184248530829456061755811587457, 19.687096100105697721076566220304, 20.39007679859908471292236007542, 20.89602634052943861359372942867, 21.66447417066370179251385498410

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