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Properties

Label 1-1575-1575.1348-r1-0-0
Degree $1$
Conductor $1575$
Sign $-0.976 + 0.215i$
Analytic cond. $169.257$
Root an. cond. $169.257$
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  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (0.669 + 0.743i)11-s + (0.207 − 0.978i)13-s + (0.309 + 0.951i)16-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.406 − 0.913i)22-s + (−0.207 − 0.978i)23-s + (−0.5 + 0.866i)26-s + (0.104 − 0.994i)29-s + (−0.809 + 0.587i)31-s − i·32-s + (−0.669 + 0.743i)34-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−0.587 − 0.809i)8-s + (0.669 + 0.743i)11-s + (0.207 − 0.978i)13-s + (0.309 + 0.951i)16-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.406 − 0.913i)22-s + (−0.207 − 0.978i)23-s + (−0.5 + 0.866i)26-s + (0.104 − 0.994i)29-s + (−0.809 + 0.587i)31-s − i·32-s + (−0.669 + 0.743i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(169.257\)
Root analytic conductor: \(169.257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (1:\ ),\ -0.976 + 0.215i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01862482440 - 0.1710116959i\)
\(L(\frac12)\) \(\approx\) \(0.01862482440 - 0.1710116959i\)
\(L(1)\) \(\approx\) \(0.6545236649 - 0.1080338968i\)
\(L(1)\) \(\approx\) \(0.6545236649 - 0.1080338968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.207 + 0.978i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.994 - 0.104i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.207 - 0.978i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.406 + 0.913i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.994 + 0.104i)T \)
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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

−20.54059452584565311993103688267, −19.59268980969694471206940536316, −19.34239023871110061610824638609, −18.466535624634056516115443130162, −17.728399752451491992910291808580, −16.95167030667104947193710164904, −16.41117057458236778254402181668, −15.6966086200732414426317246082, −14.779169048447388019998429586138, −14.19129468599141713330508473924, −13.27720890265113974610375210820, −12.14448515548918164618656521160, −11.3531410031435539636523128257, −10.88584944101659177244680325448, −9.842262226136650247436890265602, −9.0730719434180578512370745824, −8.617153027927090101369226538244, −7.58246147140862831019675401836, −6.854596587414337150263810385, −6.08457004629094650558842238187, −5.32393043334887014458932662729, −4.0374689631853624347326202728, −3.10871676197267326989299059322, −1.86781484727194063326189071055, −1.18561179839287628242536375187, 0.04808463800985686251237724680, 1.087482176607492940370092568140, 1.98332830956001506407630950211, 3.00507610342214392313379352120, 3.79402390713251127506274833788, 4.9507307076267827956363901797, 6.08299313410405012098007686327, 6.84245024747774142939132334285, 7.74366083679003855185413754210, 8.30729184109959404788912079617, 9.316297813343258699301502510935, 9.9359925030117441733993728394, 10.58596996486211187383670174422, 11.52167730006427737105824393359, 12.261129448048498164727251084, 12.762450277078012929545962247220, 13.97334023535662947005082642771, 14.79619874290110120224593373658, 15.63419680961721143504937137026, 16.31131540788737220335719585800, 17.12606286881433419698152766935, 17.70131952334987848572976013043, 18.5080689604109435923466504096, 19.01930574961676296250404661444, 20.05670818736074458490115707100

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