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Properties

Label 1-407-407.157-r0-0-0
Degree $1$
Conductor $407$
Sign $0.982 + 0.185i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.0348 + 0.999i)2-s + (0.961 − 0.275i)3-s + (−0.997 + 0.0697i)4-s + (−0.882 + 0.469i)5-s + (0.309 + 0.951i)6-s + (0.438 − 0.898i)7-s + (−0.104 − 0.994i)8-s + (0.848 − 0.529i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.374 − 0.927i)13-s + (0.913 + 0.406i)14-s + (−0.719 + 0.694i)15-s + (0.990 − 0.139i)16-s + (−0.374 + 0.927i)17-s + (0.559 + 0.829i)18-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (0.961 − 0.275i)3-s + (−0.997 + 0.0697i)4-s + (−0.882 + 0.469i)5-s + (0.309 + 0.951i)6-s + (0.438 − 0.898i)7-s + (−0.104 − 0.994i)8-s + (0.848 − 0.529i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.374 − 0.927i)13-s + (0.913 + 0.406i)14-s + (−0.719 + 0.694i)15-s + (0.990 − 0.139i)16-s + (−0.374 + 0.927i)17-s + (0.559 + 0.829i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.448359861 + 0.1354306140i\)
\(L(\frac12)\) \(\approx\) \(1.448359861 + 0.1354306140i\)
\(L(1)\) \(\approx\) \(1.179094753 + 0.2796714700i\)
\(L(1)\) \(\approx\) \(1.179094753 + 0.2796714700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
3 \( 1 + (0.961 - 0.275i)T \)
5 \( 1 + (-0.882 + 0.469i)T \)
7 \( 1 + (0.438 - 0.898i)T \)
13 \( 1 + (-0.374 - 0.927i)T \)
17 \( 1 + (-0.374 + 0.927i)T \)
19 \( 1 + (0.961 - 0.275i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.882 - 0.469i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.0348 - 0.999i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (-0.374 + 0.927i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.978 + 0.207i)T \)
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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

−24.289555471011523624273663858386, −23.42052218403414273362527703393, −22.17897293418332218944064487868, −21.57144153453100408214419223827, −20.66082179535498677449896112596, −20.08323205760259827348517317826, −19.20907670646561478439266612218, −18.65454411300993384559668855117, −17.60836653608507475374805838424, −16.1045560189032664928775583639, −15.49921545660100964408489945986, −14.28808373730000885344520477488, −13.809361953945459923747350497135, −12.42774177941298747400960484259, −11.93333899204750162180417020902, −10.997329151629307950995301041141, −9.64150027527481005731025610753, −9.039297113188398899222607212295, −8.27918864856400942075822682623, −7.30712384720824439278931221046, −5.22876051848006942405076666527, −4.5285906630489178232068779201, −3.49851713643632690620780536336, −2.53067727595534199716057770981, −1.430375208864977282628941221958, 0.864018436779759989407652993881, 2.82155269773082539302129831727, 3.91937831380942956792517456204, 4.59549906848626731902100899025, 6.25065225153039941604467165853, 7.24613830727379897684834321879, 7.8776446972481425951509391225, 8.42411862171861446639630129656, 9.783150624057729819003363492414, 10.6507286673989674727812855401, 12.14852259549855514291564591182, 13.12711058687834659255741874784, 13.98827211282183008265609987678, 14.73269090502970560268172231891, 15.36770220498142399629399251273, 16.212642390360846402316346932660, 17.45572021219682006536856334485, 18.12957775770240264959802169272, 19.15529734536294328904874036984, 19.862503459151475443287247689024, 20.695192460210021786258613297506, 22.04915945386921154119143414875, 22.81282395885473851758502530003, 23.8496420407509769731277304686, 24.219181129703115166053543992901

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