Properties

Label 2-2475-1.1-c3-0-139
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.54·2-s + 22.7·4-s − 17.7·7-s + 81.4·8-s − 11·11-s + 53.5·13-s − 98.2·14-s + 269.·16-s − 112.·17-s − 1.24·19-s − 60.9·22-s + 78.4·23-s + 296.·26-s − 402.·28-s + 174.·29-s + 82.5·31-s + 842.·32-s − 622.·34-s + 149.·37-s − 6.89·38-s + 414.·41-s + 182.·43-s − 249.·44-s + 434.·46-s + 438.·47-s − 28.5·49-s + 1.21e3·52-s + ⋯
L(s)  = 1  + 1.95·2-s + 2.83·4-s − 0.957·7-s + 3.60·8-s − 0.301·11-s + 1.14·13-s − 1.87·14-s + 4.21·16-s − 1.60·17-s − 0.0150·19-s − 0.590·22-s + 0.710·23-s + 2.23·26-s − 2.71·28-s + 1.11·29-s + 0.478·31-s + 4.65·32-s − 3.14·34-s + 0.662·37-s − 0.0294·38-s + 1.57·41-s + 0.647·43-s − 0.855·44-s + 1.39·46-s + 1.36·47-s − 0.0833·49-s + 3.24·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(9.055731387\)
\(L(\frac12)\) \(\approx\) \(9.055731387\)
\(L(\frac{5}{2})\) 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 \( 1 + 11T \)
good2 \( 1 - 5.54T + 8T^{2} \)
7 \( 1 + 17.7T + 343T^{2} \)
13 \( 1 - 53.5T + 2.19e3T^{2} \)
17 \( 1 + 112.T + 4.91e3T^{2} \)
19 \( 1 + 1.24T + 6.85e3T^{2} \)
23 \( 1 - 78.4T + 1.21e4T^{2} \)
29 \( 1 - 174.T + 2.43e4T^{2} \)
31 \( 1 - 82.5T + 2.97e4T^{2} \)
37 \( 1 - 149.T + 5.06e4T^{2} \)
41 \( 1 - 414.T + 6.89e4T^{2} \)
43 \( 1 - 182.T + 7.95e4T^{2} \)
47 \( 1 - 438.T + 1.03e5T^{2} \)
53 \( 1 - 490.T + 1.48e5T^{2} \)
59 \( 1 + 6.02T + 2.05e5T^{2} \)
61 \( 1 - 434.T + 2.26e5T^{2} \)
67 \( 1 + 935.T + 3.00e5T^{2} \)
71 \( 1 + 510.T + 3.57e5T^{2} \)
73 \( 1 - 1.04e3T + 3.89e5T^{2} \)
79 \( 1 - 226.T + 4.93e5T^{2} \)
83 \( 1 + 1.18e3T + 5.71e5T^{2} \)
89 \( 1 - 1.44e3T + 7.04e5T^{2} \)
97 \( 1 + 825.T + 9.12e5T^{2} \)
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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

−8.422932280526224686736417388936, −7.36685376359466594804795539578, −6.62740273805025938033039290519, −6.19222564393882483692647236157, −5.45331660493585234174699420418, −4.40735672365152337784015130397, −3.96779362548733993050201171160, −2.90829954985456003327152661939, −2.41603670004645563111512850544, −1.00608632849520257233458555231, 1.00608632849520257233458555231, 2.41603670004645563111512850544, 2.90829954985456003327152661939, 3.96779362548733993050201171160, 4.40735672365152337784015130397, 5.45331660493585234174699420418, 6.19222564393882483692647236157, 6.62740273805025938033039290519, 7.36685376359466594804795539578, 8.422932280526224686736417388936

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