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Properties

Label 2-1875-75.14-c0-0-5
Degree $2$
Conductor $1875$
Sign $0.125 + 0.992i$
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  + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s − 1.61i·7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)12-s + (−0.363 + 0.5i)13-s + (−0.809 − 0.587i)16-s + (−0.190 − 0.587i)19-s + (−0.500 + 1.53i)21-s + (−0.587 − 0.809i)27-s + (1.53 + 0.500i)28-s + (−0.5 − 1.53i)31-s + (−0.809 + 0.587i)36-s + (0.363 − 0.5i)37-s + (0.5 − 0.363i)39-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s − 1.61i·7-s + (0.809 + 0.587i)9-s + (0.587 − 0.809i)12-s + (−0.363 + 0.5i)13-s + (−0.809 − 0.587i)16-s + (−0.190 − 0.587i)19-s + (−0.500 + 1.53i)21-s + (−0.587 − 0.809i)27-s + (1.53 + 0.500i)28-s + (−0.5 − 1.53i)31-s + (−0.809 + 0.587i)36-s + (0.363 − 0.5i)37-s + (0.5 − 0.363i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $0.125 + 0.992i$
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: \((2,\ 1875,\ (\ :0),\ 0.125 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5871345755\)
\(L(\frac12)\) \(\approx\) \(0.5871345755\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.365920373585795876036691268426, −8.216283603394518115187020384169, −7.41277123977758017890516185888, −7.10791982266121177053863317456, −6.22660034382757938746252896928, −4.98219259889015078306362383958, −4.30294033061788434787064632882, −3.64937128499459053760774110977, −2.13895310118408728600773584926, −0.52274092829535684105010648074, 1.39039164689954388702843279640, 2.63567842624448717788632772327, 4.03248721035549088317229250563, 5.22047300821803077611711690576, 5.35803071601132460221989404100, 6.19605544687347089455474571402, 6.90804973867259459458383706555, 8.262103125316866479524380966271, 8.966465692769909698149116067665, 9.752216184341340294364345626301

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