Properties

Label 2-2475-1.1-c1-0-19
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·2-s + 0.438·4-s − 7-s + 2.43·8-s + 11-s + 4.56·13-s + 1.56·14-s − 4.68·16-s + 5.56·17-s + 3·19-s − 1.56·22-s + 2.43·23-s − 7.12·26-s − 0.438·28-s + 3.12·29-s − 7.68·31-s + 2.43·32-s − 8.68·34-s − 9.80·37-s − 4.68·38-s − 4.68·41-s + 7.68·43-s + 0.438·44-s − 3.80·46-s + 9.56·47-s − 6·49-s + 1.99·52-s + ⋯
L(s)  = 1  − 1.10·2-s + 0.219·4-s − 0.377·7-s + 0.862·8-s + 0.301·11-s + 1.26·13-s + 0.417·14-s − 1.17·16-s + 1.34·17-s + 0.688·19-s − 0.332·22-s + 0.508·23-s − 1.39·26-s − 0.0828·28-s + 0.579·29-s − 1.38·31-s + 0.431·32-s − 1.48·34-s − 1.61·37-s − 0.759·38-s − 0.731·41-s + 1.17·43-s + 0.0660·44-s − 0.561·46-s + 1.39·47-s − 0.857·49-s + 0.277·52-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9991346940\)
\(L(\frac12)\) \(\approx\) \(0.9991346940\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + 1.56T + 2T^{2} \)
7 \( 1 + T + 7T^{2} \)
13 \( 1 - 4.56T + 13T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 + 9.80T + 37T^{2} \)
41 \( 1 + 4.68T + 41T^{2} \)
43 \( 1 - 7.68T + 43T^{2} \)
47 \( 1 - 9.56T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + 6.43T + 59T^{2} \)
61 \( 1 + 3.43T + 61T^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 0.246T + 73T^{2} \)
79 \( 1 + 3.31T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 - 6.12T + 97T^{2} \)
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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

−9.027994156606032544069874377355, −8.317427360164500719333675615895, −7.53549529188499791817938419355, −6.91381696389274490866972809032, −5.89337302097861147499773196989, −5.12127610624148283958922911689, −3.91690925270182544016244310188, −3.21110396665105094870106843254, −1.68082303157079390627616932394, −0.810047830414422836938202944158, 0.810047830414422836938202944158, 1.68082303157079390627616932394, 3.21110396665105094870106843254, 3.91690925270182544016244310188, 5.12127610624148283958922911689, 5.89337302097861147499773196989, 6.91381696389274490866972809032, 7.53549529188499791817938419355, 8.317427360164500719333675615895, 9.027994156606032544069874377355

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