Properties

Label 1-403-403.292-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.303 + 0.952i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (0.207 + 0.978i)2-s + (0.978 + 0.207i)3-s + (−0.913 + 0.406i)4-s + (0.866 + 0.5i)5-s + i·6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.913 + 0.406i)9-s + (−0.309 + 0.951i)10-s + (−0.587 + 0.809i)11-s + (−0.978 + 0.207i)12-s + (0.913 + 0.406i)14-s + (0.743 + 0.669i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + (−0.207 + 0.978i)18-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (0.978 + 0.207i)3-s + (−0.913 + 0.406i)4-s + (0.866 + 0.5i)5-s + i·6-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (0.913 + 0.406i)9-s + (−0.309 + 0.951i)10-s + (−0.587 + 0.809i)11-s + (−0.978 + 0.207i)12-s + (0.913 + 0.406i)14-s + (0.743 + 0.669i)15-s + (0.669 − 0.743i)16-s + (−0.809 + 0.587i)17-s + (−0.207 + 0.978i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.279072674 + 1.749356964i\)
\(L(\frac12)\) \(\approx\) \(1.279072674 + 1.749356964i\)
\(L(1)\) \(\approx\) \(1.330527793 + 1.010139163i\)
\(L(1)\) \(\approx\) \(1.330527793 + 1.010139163i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
3 \( 1 + (0.978 + 0.207i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 + (-0.587 + 0.809i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.406 - 0.913i)T \)
73 \( 1 + (-0.406 - 0.913i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (0.994 + 0.104i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−24.29226150534260697928938572152, −23.22575970331451108125936257244, −21.77285335596325315576025983223, −21.395295713598561000692252527506, −20.766073642621897484150656553, −19.91202480050890777962857038184, −18.90515848948685009619730848008, −18.29177142091130639118020992355, −17.50992915001272518724041777798, −16.05913611202367277257818672068, −14.84468915471456898273540161696, −14.215528682327288449787089965088, −13.183934174217673756244691381206, −12.850198719402400717656638631434, −11.62286449505123644589574312972, −10.56327643879604576224757157908, −9.55932587985066903387918150157, −8.67773349537741301708639975377, −8.30459156498993466732774615456, −6.42612141414714343728681100275, −5.25026164936449481723126765937, −4.40455418978208005715440567503, −2.865574915552813433180822773865, −2.31629061695408908053629405724, −1.214771251465230046769716840274, 1.76509269077392502021562260166, 2.98756029306178651787170767311, 4.2899173058161928441423773325, 4.96616028197422403996685150560, 6.47318116606362069547569154153, 7.21244805142820372443875128053, 8.122856175968023563476556755967, 9.05665277988411835804467137136, 10.037036836427874457054930756375, 10.78174694731157509982486961699, 12.73671314242565583507550972201, 13.44554772467557569606410666644, 14.0416056353085003658318502065, 15.05884497749689127249345394898, 15.33743003568089140008672950887, 16.8249212316557673654593196175, 17.515639828259369371410696640753, 18.27524329717136115806475975897, 19.32174908607917412118303182728, 20.47372445501078931113476416124, 21.30154484869515301531164056730, 21.90160738242709797388727160728, 23.09903565217428891862883308097, 23.8385589491782132183703448316, 24.808283803811489159505259223014

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