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Properties

Label 2-1875-75.41-c0-0-1
Degree $2$
Conductor $1875$
Sign $-0.992 - 0.125i$
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 − 1.61·7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)19-s + (−0.500 − 1.53i)21-s + (−0.809 − 0.587i)27-s + (−0.500 − 1.53i)28-s + (−0.5 + 1.53i)31-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (−0.5 − 0.363i)39-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s − 1.61·7-s + (−0.809 + 0.587i)9-s + (−0.809 + 0.587i)12-s + (−0.5 + 0.363i)13-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)19-s + (−0.500 − 1.53i)21-s + (−0.809 − 0.587i)27-s + (−0.500 − 1.53i)28-s + (−0.5 + 1.53i)31-s + (−0.809 − 0.587i)36-s + (−0.5 + 0.363i)37-s + (−0.5 − 0.363i)39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7851423184\)
\(L(\frac12)\) \(\approx\) \(0.7851423184\)
\(L(1)\) 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 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
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 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
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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

−9.741702277942218722982714860478, −8.989828084828508103180308738143, −8.512861049853999210905156922943, −7.30494656490960397105812764601, −6.82755600630507913029652339623, −5.80451583630292919154897675188, −4.72872666781580916486029520623, −3.78845628100537775842168761987, −3.15809640020525841237836572774, −2.43884938936340371563717982119, 0.51041222902844062686983641659, 1.98669727052638960055318936484, 2.85343332041804437076657008102, 3.81955561652559520443276044335, 5.36348899730402872152044130469, 6.00832532584444207354133529298, 6.63699365430489058252042970738, 7.29283693784706428632335156504, 8.142008094478279935729997093348, 9.426258998544724202605679102358

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