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Properties

Label 2-1875-75.14-c0-0-7
Degree $2$
Conductor $1875$
Sign $-0.728 + 0.684i$
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  + (1.30 − 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−1.30 − 0.951i)12-s + (0.190 + 0.587i)17-s − 1.61·18-s + (−0.5 − 1.53i)19-s + (−0.5 + 0.363i)23-s − 24-s + (−0.809 + 0.587i)27-s + (0.190 + 0.587i)31-s + 0.999·32-s + ⋯
L(s)  = 1  + (1.30 − 0.951i)2-s + (0.309 − 0.951i)3-s + (0.500 − 1.53i)4-s + (−0.499 − 1.53i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−1.30 − 0.951i)12-s + (0.190 + 0.587i)17-s − 1.61·18-s + (−0.5 − 1.53i)19-s + (−0.5 + 0.363i)23-s − 24-s + (−0.809 + 0.587i)27-s + (0.190 + 0.587i)31-s + 0.999·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.728 + 0.684i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ -0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.439268810\)
\(L(\frac12)\) \(\approx\) \(2.439268810\)
\(L(1)\) 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 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−9.130766226633422229361870070792, −8.330698593603980360063380107157, −7.41952141476593342594447998135, −6.49649984618601743818494395341, −5.84747670414960408123842435394, −4.90479148422399580083511592236, −4.00190767907208405345901527751, −3.04025746137717714911892571417, −2.33966117515584781186243765963, −1.29822604451671053704021253931, 2.33916440598841405585886715746, 3.50865132202479793187701128798, 4.01188343431428089732522480623, 4.88484688508733627492178741964, 5.59113532121482175128416797775, 6.25127413406143267428273026622, 7.23801603077885422340971636267, 8.073350166389081524433260189099, 8.655045134248462617384608958610, 9.879984438546558012790760037413

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