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Properties

Label 2-2475-2475.1429-c0-0-1
Degree $2$
Conductor $2475$
Sign $0.475 + 0.879i$
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  + (0.913 + 0.406i)3-s + (0.104 − 0.994i)4-s + (0.5 − 0.866i)5-s + (0.669 + 0.743i)9-s + (−0.669 − 0.743i)11-s + (0.499 − 0.866i)12-s + (0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.809 − 0.587i)20-s + (−0.244 − 1.14i)23-s + (−0.499 − 0.866i)25-s + (0.309 + 0.951i)27-s + (1.78 + 0.795i)31-s + (−0.309 − 0.951i)33-s + (0.809 − 0.587i)36-s + (−1.64 + 0.535i)37-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)3-s + (0.104 − 0.994i)4-s + (0.5 − 0.866i)5-s + (0.669 + 0.743i)9-s + (−0.669 − 0.743i)11-s + (0.499 − 0.866i)12-s + (0.809 − 0.587i)15-s + (−0.978 − 0.207i)16-s + (−0.809 − 0.587i)20-s + (−0.244 − 1.14i)23-s + (−0.499 − 0.866i)25-s + (0.309 + 0.951i)27-s + (1.78 + 0.795i)31-s + (−0.309 − 0.951i)33-s + (0.809 − 0.587i)36-s + (−1.64 + 0.535i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.475 + 0.879i$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ 0.475 + 0.879i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.761545497\)
\(L(\frac12)\) \(\approx\) \(1.761545497\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
good2 \( 1 + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.244 + 1.14i)T + (-0.913 + 0.406i)T^{2} \)
29 \( 1 + (-0.669 + 0.743i)T^{2} \)
31 \( 1 + (-1.78 - 0.795i)T + (0.669 + 0.743i)T^{2} \)
37 \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.104 + 0.994i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.704 - 1.58i)T + (-0.669 + 0.743i)T^{2} \)
53 \( 1 + (0.244 - 0.336i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.895 + 0.994i)T + (-0.104 - 0.994i)T^{2} \)
61 \( 1 + (0.104 - 0.994i)T^{2} \)
67 \( 1 + (0.330 - 0.743i)T + (-0.669 - 0.743i)T^{2} \)
71 \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.669 + 0.743i)T^{2} \)
83 \( 1 + (-0.978 + 0.207i)T^{2} \)
89 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.169 - 0.379i)T + (-0.669 + 0.743i)T^{2} \)
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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

−8.812223949411254388720547721273, −8.597101947114903469353018153002, −7.65192391287549897915198241235, −6.52423062891948727127584052957, −5.79907866164173594576959742468, −4.89560041681209037557857813887, −4.45938848061859135439878018513, −3.06365932284871284734088029317, −2.21118921788441715838188388282, −1.10776813058245870464194479893, 1.93858097232435537373488177245, 2.55248516265194544987151011413, 3.39527119844195110255781678788, 4.12628911477199775636204377201, 5.37241055015545753410799606394, 6.50328491326030672841300764040, 7.16987787693775937126046264265, 7.60587528800752076672427247475, 8.383586120590837623036155345791, 9.118244087294033210341660914369

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