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Properties

Label 1-643-643.21-r1-0-0
Degree $1$
Conductor $643$
Sign $-0.426 + 0.904i$
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  + (0.999 + 0.0293i)2-s + (0.412 − 0.910i)3-s + (0.998 + 0.0586i)4-s + (−0.218 + 0.975i)5-s + (0.439 − 0.898i)6-s + (−0.673 − 0.739i)7-s + (0.996 + 0.0879i)8-s + (−0.658 − 0.752i)9-s + (−0.246 + 0.969i)10-s + (−0.349 + 0.936i)11-s + (0.465 − 0.884i)12-s + (0.935 + 0.354i)13-s + (−0.651 − 0.758i)14-s + (0.798 + 0.601i)15-s + (0.993 + 0.117i)16-s + (−0.873 + 0.487i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0293i)2-s + (0.412 − 0.910i)3-s + (0.998 + 0.0586i)4-s + (−0.218 + 0.975i)5-s + (0.439 − 0.898i)6-s + (−0.673 − 0.739i)7-s + (0.996 + 0.0879i)8-s + (−0.658 − 0.752i)9-s + (−0.246 + 0.969i)10-s + (−0.349 + 0.936i)11-s + (0.465 − 0.884i)12-s + (0.935 + 0.354i)13-s + (−0.651 − 0.758i)14-s + (0.798 + 0.601i)15-s + (0.993 + 0.117i)16-s + (−0.873 + 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(69.0999\)
Root analytic conductor: \(69.0999\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (1:\ ),\ -0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9132137184 + 1.441142742i\)
\(L(\frac12)\) \(\approx\) \(0.9132137184 + 1.441142742i\)
\(L(1)\) \(\approx\) \(1.621625684 + 0.01718906812i\)
\(L(1)\) \(\approx\) \(1.621625684 + 0.01718906812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (0.999 + 0.0293i)T \)
3 \( 1 + (0.412 - 0.910i)T \)
5 \( 1 + (-0.218 + 0.975i)T \)
7 \( 1 + (-0.673 - 0.739i)T \)
11 \( 1 + (-0.349 + 0.936i)T \)
13 \( 1 + (0.935 + 0.354i)T \)
17 \( 1 + (-0.873 + 0.487i)T \)
19 \( 1 + (-0.786 + 0.617i)T \)
23 \( 1 + (-0.628 + 0.777i)T \)
29 \( 1 + (-0.986 + 0.165i)T \)
31 \( 1 + (-0.920 - 0.390i)T \)
37 \( 1 + (-0.199 - 0.979i)T \)
41 \( 1 + (0.970 + 0.242i)T \)
43 \( 1 + (0.815 - 0.578i)T \)
47 \( 1 + (-0.976 + 0.213i)T \)
53 \( 1 + (-0.340 + 0.940i)T \)
59 \( 1 + (-0.995 - 0.0977i)T \)
61 \( 1 + (-0.508 + 0.861i)T \)
67 \( 1 + (0.516 + 0.856i)T \)
71 \( 1 + (0.701 - 0.712i)T \)
73 \( 1 + (0.999 + 0.00978i)T \)
79 \( 1 + (0.265 - 0.964i)T \)
83 \( 1 + (-0.920 + 0.390i)T \)
89 \( 1 + (0.525 + 0.850i)T \)
97 \( 1 + (0.996 + 0.0782i)T \)
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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

−22.34920675464423819394342881029, −21.521194890744222339943492185, −21.00087195648905556117464147106, −20.14115410741920041339722462945, −19.60524345948986176276317456841, −18.54371441929240721456948728595, −16.98076610009958853870418943240, −16.114861602349214226987281800485, −15.842115226088957515645866417100, −15.1105474648892049371686594820, −13.97866463319534816865853471512, −13.17598531881152432944930934557, −12.637103231513669679595528161030, −11.34558267769719063283494037080, −10.87302409572180759917449157872, −9.56505361811155879928272780435, −8.714530322920351733832299210854, −8.03702028013558860234052857534, −6.40632548322559395559389290397, −5.621104736931202935587515214008, −4.8131300504653662465135569988, −3.88692519125247231481021735692, −3.07742400838868357633662698675, −2.06631163960203875488275066222, −0.22661085747366057745019020391, 1.67055934182155329796274454691, 2.41768691167274207455662141800, 3.654692938808064325216482660017, 4.04147067683400016660881759887, 5.90461400312808469155617680542, 6.45910091393484386403450733206, 7.31432335192553859998596241551, 7.81627709700537832786079793361, 9.355059583848778351063928608, 10.63049947308465094220474264410, 11.16399357191798593055030445352, 12.35483074092209401104100610509, 13.0052042614303329740421028473, 13.701666103137537737944627507640, 14.47307166566347406812439316201, 15.20434260382088125488358834681, 16.044275154785178937300160774488, 17.206981683438741585434567723837, 18.12397921767169517174231973263, 19.05782777337230196396376948745, 19.758999036708938276807810911737, 20.39369563775889107308070859474, 21.352974479562609163454551107572, 22.47076801409826345824558452092, 23.07125625757835326519227366982

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