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Properties

Label 1-6300-6300.1067-r0-0-0
Degree $1$
Conductor $6300$
Sign $-0.837 + 0.546i$
Analytic cond. $29.2570$
Root an. cond. $29.2570$
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  + (−0.809 + 0.587i)11-s + (−0.994 − 0.104i)13-s + (0.743 + 0.669i)17-s + (0.978 + 0.207i)19-s + (0.587 + 0.809i)23-s + (0.669 + 0.743i)29-s + (−0.978 − 0.207i)31-s + (0.406 + 0.913i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (0.207 + 0.978i)47-s + (−0.207 − 0.978i)53-s + (0.913 − 0.406i)59-s + (−0.913 − 0.406i)61-s + (−0.207 + 0.978i)67-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)11-s + (−0.994 − 0.104i)13-s + (0.743 + 0.669i)17-s + (0.978 + 0.207i)19-s + (0.587 + 0.809i)23-s + (0.669 + 0.743i)29-s + (−0.978 − 0.207i)31-s + (0.406 + 0.913i)37-s + (−0.104 + 0.994i)41-s + (−0.866 − 0.5i)43-s + (0.207 + 0.978i)47-s + (−0.207 − 0.978i)53-s + (0.913 − 0.406i)59-s + (−0.913 − 0.406i)61-s + (−0.207 + 0.978i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.837 + 0.546i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ -0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2470720909 + 0.8304346676i\)
\(L(\frac12)\) \(\approx\) \(0.2470720909 + 0.8304346676i\)
\(L(1)\) \(\approx\) \(0.8974501155 + 0.1807195539i\)
\(L(1)\) \(\approx\) \(0.8974501155 + 0.1807195539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.207 + 0.978i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.406 + 0.913i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.743 - 0.669i)T \)
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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

−17.30245988869537777622042695283, −16.63164679160038245002330466750, −16.14088946843228256895427931572, −15.4635449800510343749135057331, −14.688198881763226046781784434321, −14.12345477667850494446462761147, −13.483720800314147313420508916859, −12.78335724818153395434591661497, −12.05510128922446797409364487516, −11.567012483126367954592415821375, −10.64958392323860464340020189849, −10.183786542621796736227654039214, −9.37469840029452587131381628746, −8.815006986845239850622440951850, −7.84927532773232147658047087058, −7.436263243906186127728809690183, −6.71826530899958163583279875871, −5.70279478203049005889743005612, −5.22593970122057366810974847911, −4.568447152172423552349393049761, −3.55026733012619705924597810243, −2.82327919838741122781973408219, −2.29081007216316271776580861469, −1.086098475809155130172665517089, −0.23403532614639414037462922137, 1.15907263320744782358056768094, 1.868440516038702300867658917330, 2.89608505317791056632308666305, 3.3283968470134676352144354747, 4.405394052864047430347989335460, 5.15639452657160810458904295359, 5.52335065474074913541915637263, 6.569625518412754866605035104101, 7.33435357638872262499305431729, 7.77670154450666443463084011109, 8.49372168881039432456474884898, 9.52536284089027373832723692145, 9.89181211777378753345947080503, 10.53349228644743885207467430875, 11.396950382047654007881599199146, 12.02024212598115186296453758099, 12.733055105385505919411717873533, 13.16291945672825994504420591016, 14.06355660472758014935175586423, 14.74751988651706811367445409731, 15.16173940890691362055583990967, 15.995729086183747044702992082873, 16.56337750943966649385175889015, 17.31134937359703124562267136903, 17.83396452921070387092236701766

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