Properties

Label 2-95e2-1.1-c1-0-189
Degree $2$
Conductor $9025$
Sign $1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
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  − 1.19·2-s + 2.27·3-s − 0.574·4-s − 2.71·6-s + 2.19·7-s + 3.07·8-s + 2.16·9-s − 2.82·11-s − 1.30·12-s + 4.08·13-s − 2.62·14-s − 2.52·16-s − 0.382·17-s − 2.58·18-s + 4.99·21-s + 3.37·22-s + 2.97·23-s + 6.98·24-s − 4.87·26-s − 1.89·27-s − 1.26·28-s − 8.57·29-s + 4.69·31-s − 3.13·32-s − 6.42·33-s + 0.456·34-s − 1.24·36-s + ⋯
L(s)  = 1  − 0.844·2-s + 1.31·3-s − 0.287·4-s − 1.10·6-s + 0.830·7-s + 1.08·8-s + 0.722·9-s − 0.852·11-s − 0.376·12-s + 1.13·13-s − 0.700·14-s − 0.630·16-s − 0.0928·17-s − 0.609·18-s + 1.08·21-s + 0.719·22-s + 0.620·23-s + 1.42·24-s − 0.955·26-s − 0.364·27-s − 0.238·28-s − 1.59·29-s + 0.842·31-s − 0.554·32-s − 1.11·33-s + 0.0783·34-s − 0.207·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.058401308\)
\(L(\frac12)\) \(\approx\) \(2.058401308\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + 1.19T + 2T^{2} \)
3 \( 1 - 2.27T + 3T^{2} \)
7 \( 1 - 2.19T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 + 0.382T + 17T^{2} \)
23 \( 1 - 2.97T + 23T^{2} \)
29 \( 1 + 8.57T + 29T^{2} \)
31 \( 1 - 4.69T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 3.13T + 41T^{2} \)
43 \( 1 + 2.00T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 - 7.47T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 5.97T + 61T^{2} \)
67 \( 1 - 1.04T + 67T^{2} \)
71 \( 1 - 7.82T + 71T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 5.55T + 83T^{2} \)
89 \( 1 - 6.56T + 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−8.008969145001365683750461245006, −7.45619918011821325380668723168, −6.72274830172079890297056163611, −5.44199500678946979352807071322, −5.04157642126148041907309082809, −3.91423730635801127743564493906, −3.55611887689334075267309444570, −2.36473973259520842542601089614, −1.79334718723796537714382587646, −0.75950054406468337250790481063, 0.75950054406468337250790481063, 1.79334718723796537714382587646, 2.36473973259520842542601089614, 3.55611887689334075267309444570, 3.91423730635801127743564493906, 5.04157642126148041907309082809, 5.44199500678946979352807071322, 6.72274830172079890297056163611, 7.45619918011821325380668723168, 8.008969145001365683750461245006

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