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Properties

Label 2-1875-75.11-c0-0-5
Degree $2$
Conductor $1875$
Sign $-0.187 + 0.982i$
Analytic cond. $0.935746$
Root an. cond. $0.967340$
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  + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s − 0.618·7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)12-s + (−1.30 − 0.951i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)19-s + (0.190 − 0.587i)21-s + (0.809 − 0.587i)27-s + (−0.190 + 0.587i)28-s + (0.190 + 0.587i)31-s + (−0.809 + 0.587i)36-s + (−1.30 − 0.951i)37-s + (1.30 − 0.951i)39-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s − 0.618·7-s + (−0.809 − 0.587i)9-s + (0.809 + 0.587i)12-s + (−1.30 − 0.951i)13-s + (−0.809 − 0.587i)16-s + (−0.5 − 1.53i)19-s + (0.190 − 0.587i)21-s + (0.809 − 0.587i)27-s + (−0.190 + 0.587i)28-s + (0.190 + 0.587i)31-s + (−0.809 + 0.587i)36-s + (−1.30 − 0.951i)37-s + (1.30 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5968429062\)
\(L(\frac12)\) \(\approx\) \(0.5968429062\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.374803633825659262928441660461, −8.788329428616469468951509252465, −7.44840519487756075923701422217, −6.69378336287375083059100299812, −5.86142857170373497077689649083, −5.12400355522811983823010592885, −4.52961968931655394601310836139, −3.20444241165882031220470250345, −2.38509160433393066326123990110, −0.41841919244613947414339120230, 1.83638900186315710694170740351, 2.65017991113359033453167492939, 3.70371992549256929582244855287, 4.73333603550058432482249038388, 5.92778360303638242317063924776, 6.62384302949570444329833227643, 7.26495465030481989199922229551, 7.902298746615295880837978090709, 8.652809425617593363117494640115, 9.566835087552867517029059838018

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