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Properties

Label 1-6300-6300.2083-r0-0-0
Degree $1$
Conductor $6300$
Sign $-0.961 - 0.275i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.961 - 0.275i$
Analytic conductor: \(29.2570\)
Root analytic conductor: \(29.2570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (2083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6300,\ (0:\ ),\ -0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06998242310 - 0.4989342973i\)
\(L(\frac12)\) \(\approx\) \(0.06998242310 - 0.4989342973i\)
\(L(1)\) \(\approx\) \(0.8819171645 - 0.1179006339i\)
\(L(1)\) \(\approx\) \(0.8819171645 - 0.1179006339i\)

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

−17.607822904263007038459956145213, −17.30386321028961755244614453528, −16.84863231118291371433169075709, −15.92865924546076269088516685789, −15.200497444974762183644055088310, −14.663546468381674476672851397233, −14.23366994047376768874696366217, −13.202133421698345675213643857315, −12.682556467783693606091116582951, −12.086078443890893296942993976116, −11.42947910105953001302895885220, −10.49751021259962138615129401408, −10.190200540925219417329808900424, −9.08742104339408721213148692237, −8.864122409487572143343704074913, −7.90228950289872451305811271269, −7.11177946799706135821662476753, −6.59445314557085186679733550460, −5.913041241932940197835611812159, −4.85815624242654947947988203605, −4.40759464733709833958655156960, −3.69059683050059164915789799543, −2.622450131650361568129270568332, −2.04391709969206175103220954142, −1.17394722411971410179968964284, 0.129815715820505177610483003295, 1.12759959345151168349853625143, 2.165590832398595049462699517321, 2.7468414673228396554154214617, 3.7135977107342239566624454899, 4.35950372578385771791406599025, 5.12846759311452904318350277516, 5.84049847541224213401277348850, 6.75208424705611013182575650357, 7.08394268676371083242945670956, 8.091899840069331711694044566291, 8.67552586769621722813637809855, 9.4695875074877085889551369250, 9.87570873802280359403468349269, 10.90787168760755553782923123843, 11.42255673310297963925601575255, 11.95492620084978606061925073621, 12.88507814926995229747243878759, 13.32379600622744404553606169950, 14.16034919831692245941562377784, 14.69834937029585796879189502331, 15.35069221194104519775173776712, 16.031894492036025625579048882888, 16.76848243903917868729112345519, 17.38726198548260688920032324276

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