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Properties

Label 1-693-693.20-r0-0-0
Degree $1$
Conductor $693$
Sign $0.686 + 0.726i$
Analytic cond. $3.21827$
Root an. cond. $3.21827$
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  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (−0.309 + 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + (−0.669 + 0.743i)29-s + (−0.913 + 0.406i)31-s + (0.5 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (−0.309 + 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (0.913 + 0.406i)16-s + (−0.809 + 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)20-s + (0.5 − 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + (−0.669 + 0.743i)29-s + (−0.913 + 0.406i)31-s + (0.5 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.686 + 0.726i$
Analytic conductor: \(3.21827\)
Root analytic conductor: \(3.21827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 693,\ (0:\ ),\ 0.686 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3402445253 + 0.1466092997i\)
\(L(\frac12)\) \(\approx\) \(0.3402445253 + 0.1466092997i\)
\(L(1)\) \(\approx\) \(0.6449159998 - 0.3526488301i\)
\(L(1)\) \(\approx\) \(0.6449159998 - 0.3526488301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−22.71352161584601203005847398117, −21.97922873153170505602604650013, −21.34680182866736100638007614962, −19.85855919682113229672921325284, −19.22442786261670318550070708600, −18.18729226644279616377738760374, −17.6949968054759656051448850016, −16.86308254214345847645516747464, −15.78725081859339092085848085243, −15.11799843511834167659826950359, −14.58375033021268921896198529727, −13.56583939542978825663233048032, −12.931274469351956203037662539437, −11.674221907285809041759506791201, −10.81348501983331331674699046324, −9.68276457371086699174914696166, −9.04394851192424152622698072030, −7.68047725857975860295527887230, −7.265540788247609837923119303554, −6.36601084629461619797059985682, −5.396625112548115260796888861275, −4.42843924640302263413063826107, −3.35481230869882067851262575215, −2.30898142288401742598798958890, −0.173456981875923231626252127870, 1.371969119322268235205734397503, 2.2154281229635994683522170673, 3.529718489895495135821533453661, 4.48755376489949352909779083324, 5.101685159758607775955854489718, 6.27037706689020212610025707123, 7.71446510212369381980444642063, 8.67625722787962700252973545902, 9.273986459993243608272057597897, 10.23981313981343793257072118778, 11.10963868610076181084258954706, 12.11022710003232186951207787152, 12.648624011379135731757404490187, 13.350976880754641659733564980764, 14.42685710440449664057603965374, 15.13682926417403050877933163286, 16.57832427023559203333843796800, 16.974008449497603802743478093543, 18.051154811208265956299865948045, 18.88173997292057403902013148914, 19.78806647653135339950540873458, 20.23267918714921543617237443854, 21.133712236800485580055864883745, 21.78431052782533959671650374674, 22.6050337594513087004762036901

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