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Properties

Label 2-2475-2475.1231-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.862 + 0.506i$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  + (0.669 − 0.743i)3-s + (0.913 + 0.406i)4-s + 5-s + (−0.104 − 0.994i)9-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (0.669 − 0.743i)15-s + (0.669 + 0.743i)16-s + (0.913 + 0.406i)20-s + (−1.08 + 1.20i)23-s + 25-s + (−0.809 − 0.587i)27-s + (0.204 − 1.94i)31-s + (−0.809 + 0.587i)33-s + (0.309 − 0.951i)36-s + (−0.309 + 0.951i)37-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)3-s + (0.913 + 0.406i)4-s + 5-s + (−0.104 − 0.994i)9-s + (−0.978 − 0.207i)11-s + (0.913 − 0.406i)12-s + (0.669 − 0.743i)15-s + (0.669 + 0.743i)16-s + (0.913 + 0.406i)20-s + (−1.08 + 1.20i)23-s + 25-s + (−0.809 − 0.587i)27-s + (0.204 − 1.94i)31-s + (−0.809 + 0.587i)33-s + (0.309 − 0.951i)36-s + (−0.309 + 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.862 + 0.506i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.862 + 0.506i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.073358766\)
\(L(\frac12)\) \(\approx\) \(2.073358766\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 - T \)
11 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (-0.913 - 0.406i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.913 + 0.406i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (1.08 - 1.20i)T + (-0.104 - 0.994i)T^{2} \)
29 \( 1 + (0.978 - 0.207i)T^{2} \)
31 \( 1 + (-0.204 + 1.94i)T + (-0.978 - 0.207i)T^{2} \)
37 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.913 + 0.406i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
53 \( 1 + (-1.58 + 1.14i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.30 - 0.278i)T + (0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (0.190 - 1.81i)T + (-0.978 - 0.207i)T^{2} \)
71 \( 1 + (1.08 - 0.786i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.204 - 1.94i)T + (-0.978 + 0.207i)T^{2} \)
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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

−8.923032370719836924253345648295, −8.075358938905140937388921131002, −7.62563023524035540555415908235, −6.80365515917810419411858770924, −6.03912913105041972034634743950, −5.49978589325612781840228255914, −3.98196529205870678825267306872, −2.94043599670227357973101431623, −2.34137854927114415756288053364, −1.50235700154777986752951322121, 1.73921521345363700866446797699, 2.48653417866548456418454662601, 3.18617292057381446651847990243, 4.53640623428566239681226528593, 5.29335996644695764895107927807, 5.98080310034885982061526301512, 6.86491131239001522554132569178, 7.69111708122400527045960485901, 8.514631126406013170738524121984, 9.272728230326095882716889329315

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