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Properties

Label 2-1875-75.29-c0-0-3
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: $\chi_{1875} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.639317199\)
\(L(\frac12)\) \(\approx\) \(1.639317199\)
\(L(1)\) 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

−9.171586111442384576176124861072, −8.612045353013506487349800719319, −7.82323844428484682680998946096, −7.25300476014599613979508261036, −6.26718398117752928618141946106, −5.81427426145748830946055043052, −4.46969760755048894813850039104, −3.02973380383215546786077691485, −2.64929377218383795263950035720, −1.73147101453333990070467208382, 1.26997386711042966714322101059, 2.61578725805436290156611837348, 3.48746360461996306931662046187, 4.45358827697622978677202684552, 5.17769829752642598801356987428, 6.26318346851229141356728130037, 7.22044902150833981226671438929, 7.65775355977819872408987381540, 8.608966205033971110824146647351, 9.806149300310393856661135668745

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