Properties

Label 2-1875-75.44-c0-0-1
Degree $2$
Conductor $1875$
Sign $0.968 + 0.248i$
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  + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + 1.61i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + (0.951 − 0.309i)27-s + (0.951 + 1.30i)28-s + (1.30 + 0.951i)31-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.190 − 0.587i)39-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + 1.61i·7-s + (−0.309 + 0.951i)9-s + (−0.951 − 0.309i)12-s + (0.587 + 0.190i)13-s + (0.309 − 0.951i)16-s + (0.5 + 0.363i)19-s + (1.30 − 0.951i)21-s + (0.951 − 0.309i)27-s + (0.951 + 1.30i)28-s + (1.30 + 0.951i)31-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.190 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.175294141\)
\(L(\frac12)\) \(\approx\) \(1.175294141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.391829573068222482994469779278, −8.470601193779764056153936188873, −7.80879718033513055432158258237, −6.69892626829503786097652054294, −6.28752317562240330201402042642, −5.52944901579529506284120101342, −4.95189029865496025224413786544, −3.13217450272737320495444632301, −2.24726181138847089410396743827, −1.37882249229111079078397906110, 1.09031273735873768404186523438, 2.83646655369922482286150962358, 3.78868687353925035687395458590, 4.27646716121372417093779211507, 5.42482274342946827408252351234, 6.41937969329933345830321219682, 6.98505175608264517104208257075, 7.78979507560133526041140296702, 8.609937632698669853413229645864, 9.705720838354120001566462437522

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