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Properties

Label 1-643-643.16-r0-0-0
Degree $1$
Conductor $643$
Sign $0.681 + 0.731i$
Analytic cond. $2.98607$
Root an. cond. $2.98607$
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  + (−0.928 − 0.372i)2-s + (0.680 + 0.732i)3-s + (0.722 + 0.691i)4-s + (0.275 + 0.961i)5-s + (−0.358 − 0.933i)6-s + (0.761 − 0.647i)7-s + (−0.412 − 0.910i)8-s + (−0.0733 + 0.997i)9-s + (0.102 − 0.994i)10-s + (0.439 − 0.898i)11-s + (−0.0146 + 0.999i)12-s + (0.863 − 0.504i)13-s + (−0.948 + 0.317i)14-s + (−0.516 + 0.856i)15-s + (0.0440 + 0.999i)16-s + (−0.189 + 0.981i)17-s + ⋯
L(s)  = 1  + (−0.928 − 0.372i)2-s + (0.680 + 0.732i)3-s + (0.722 + 0.691i)4-s + (0.275 + 0.961i)5-s + (−0.358 − 0.933i)6-s + (0.761 − 0.647i)7-s + (−0.412 − 0.910i)8-s + (−0.0733 + 0.997i)9-s + (0.102 − 0.994i)10-s + (0.439 − 0.898i)11-s + (−0.0146 + 0.999i)12-s + (0.863 − 0.504i)13-s + (−0.948 + 0.317i)14-s + (−0.516 + 0.856i)15-s + (0.0440 + 0.999i)16-s + (−0.189 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(643\)
Sign: $0.681 + 0.731i$
Analytic conductor: \(2.98607\)
Root analytic conductor: \(2.98607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{643} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 643,\ (0:\ ),\ 0.681 + 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330502907 + 0.5786055739i\)
\(L(\frac12)\) \(\approx\) \(1.330502907 + 0.5786055739i\)
\(L(1)\) \(\approx\) \(1.056381520 + 0.2339433167i\)
\(L(1)\) \(\approx\) \(1.056381520 + 0.2339433167i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad643 \( 1 \)
good2 \( 1 + (-0.928 - 0.372i)T \)
3 \( 1 + (0.680 + 0.732i)T \)
5 \( 1 + (0.275 + 0.961i)T \)
7 \( 1 + (0.761 - 0.647i)T \)
11 \( 1 + (0.439 - 0.898i)T \)
13 \( 1 + (0.863 - 0.504i)T \)
17 \( 1 + (-0.189 + 0.981i)T \)
19 \( 1 + (-0.246 + 0.969i)T \)
23 \( 1 + (0.998 - 0.0586i)T \)
29 \( 1 + (0.439 - 0.898i)T \)
31 \( 1 + (-0.516 - 0.856i)T \)
37 \( 1 + (0.491 + 0.870i)T \)
41 \( 1 + (-0.465 - 0.884i)T \)
43 \( 1 + (0.160 + 0.986i)T \)
47 \( 1 + (0.761 - 0.647i)T \)
53 \( 1 + (-0.658 - 0.752i)T \)
59 \( 1 + (0.680 - 0.732i)T \)
61 \( 1 + (0.386 + 0.922i)T \)
67 \( 1 + (-0.701 - 0.712i)T \)
71 \( 1 + (0.636 - 0.771i)T \)
73 \( 1 + (0.386 + 0.922i)T \)
79 \( 1 + (0.636 + 0.771i)T \)
83 \( 1 + (-0.516 + 0.856i)T \)
89 \( 1 + (-0.928 + 0.372i)T \)
97 \( 1 + (-0.999 + 0.0293i)T \)
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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

−23.36453011121112456595958766695, −21.63727002538329088392949657053, −20.7193548996960039280015536026, −20.276686685269898384531467136257, −19.464630568914207963554205143221, −18.50934693727354711703956937400, −17.855747487599141988265095811135, −17.33062623877211019592472817308, −16.179219511576233794263344171337, −15.383789143573445862508530042315, −14.53310564287171133654370448993, −13.7363729042850304544534875922, −12.63093276932099020917373207480, −11.8507053805166920555987733313, −10.9706578091031346172534032983, −9.39676157154243559901382680684, −9.025308901624524797446059596131, −8.45358192612778434417504397202, −7.350515572371945371200407687531, −6.6700485720603231972301181471, −5.48965185260045749118672218507, −4.53627133489582954865748073498, −2.7246681719962281550680392097, −1.760898955019347472800295181365, −1.08270677147149928032554916018, 1.31713210790244523598435089609, 2.39247008872968023362525277927, 3.498743909111463539654942903889, 3.98538867456279923457231494202, 5.76063481886207214668878614827, 6.77027961897182300898511519843, 8.04105770797456768498817201537, 8.30418300402543133240350926279, 9.48413572929580060678901572694, 10.385251126512898039440458408450, 10.88182377328868184405156344406, 11.471505960549538988154631892153, 13.10690199334946375952673403735, 13.9042931923510451799624469962, 14.84451684547486328017662384089, 15.44576417497801388002197291097, 16.65696633187423511192754813047, 17.14598206153422551899334378779, 18.20718623989545131764321178706, 18.99806957364845052521862049229, 19.59006825073943240553880309685, 20.70557897132314480890271470241, 21.05470819503722275570977442309, 21.8808172209624461293231206626, 22.717526545391483560719910529142

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