Properties

Label 2-2475-1.1-c1-0-4
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  − 2.52·2-s + 4.37·4-s − 3.46·7-s − 5.98·8-s + 11-s + 8.74·14-s + 6.37·16-s + 1.58·17-s + 4·19-s − 2.52·22-s + 0.792·23-s − 15.1·28-s − 8.74·29-s + 3.37·31-s − 4.10·32-s − 4·34-s + 1.08·37-s − 10.0·38-s − 8.74·41-s − 3.46·43-s + 4.37·44-s − 2·46-s − 6.63·47-s + 4.99·49-s + 10.0·53-s + 20.7·56-s + 22.0·58-s + ⋯
L(s)  = 1  − 1.78·2-s + 2.18·4-s − 1.30·7-s − 2.11·8-s + 0.301·11-s + 2.33·14-s + 1.59·16-s + 0.384·17-s + 0.917·19-s − 0.538·22-s + 0.165·23-s − 2.86·28-s − 1.62·29-s + 0.605·31-s − 0.726·32-s − 0.685·34-s + 0.178·37-s − 1.63·38-s − 1.36·41-s − 0.528·43-s + 0.659·44-s − 0.294·46-s − 0.967·47-s + 0.714·49-s + 1.38·53-s + 2.77·56-s + 2.89·58-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

\(L(1)\) \(\approx\) \(0.5251551977\)
\(L(\frac12)\) \(\approx\) \(0.5251551977\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.52T + 2T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.58T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 0.792T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 - 1.08T + 37T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + 6.63T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 7.37T + 59T^{2} \)
61 \( 1 + 0.744T + 61T^{2} \)
67 \( 1 + 9.30T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 - 1.25T + 79T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + 5.84T + 97T^{2} \)
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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

−9.041300363325656658920356873956, −8.322844857902938431869394498225, −7.48995684864521135684159316117, −6.87880868975869327728601818930, −6.25181260067699665293515135259, −5.30020533636251128982039308231, −3.68980322691500263352187302522, −2.93875926493938798921514204350, −1.78618455404133089448265138517, −0.59545285862512696724257669441, 0.59545285862512696724257669441, 1.78618455404133089448265138517, 2.93875926493938798921514204350, 3.68980322691500263352187302522, 5.30020533636251128982039308231, 6.25181260067699665293515135259, 6.87880868975869327728601818930, 7.48995684864521135684159316117, 8.322844857902938431869394498225, 9.041300363325656658920356873956

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