Properties

Label 2-95e2-1.1-c1-0-123
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  − 1.34·2-s − 2.87·3-s − 0.184·4-s + 3.87·6-s − 0.347·7-s + 2.94·8-s + 5.29·9-s − 2.22·11-s + 0.532·12-s + 2.57·13-s + 0.467·14-s − 3.59·16-s − 0.467·17-s − 7.12·18-s + 21-s + 3·22-s + 2.69·23-s − 8.47·24-s − 3.46·26-s − 6.59·27-s + 0.0641·28-s + 6.87·29-s + 7.10·31-s − 1.04·32-s + 6.41·33-s + 0.630·34-s − 0.977·36-s + ⋯
L(s)  = 1  − 0.952·2-s − 1.66·3-s − 0.0923·4-s + 1.58·6-s − 0.131·7-s + 1.04·8-s + 1.76·9-s − 0.671·11-s + 0.153·12-s + 0.713·13-s + 0.125·14-s − 0.899·16-s − 0.113·17-s − 1.68·18-s + 0.218·21-s + 0.639·22-s + 0.561·23-s − 1.73·24-s − 0.680·26-s − 1.26·27-s + 0.0121·28-s + 1.27·29-s + 1.27·31-s − 0.184·32-s + 1.11·33-s + 0.108·34-s − 0.162·36-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

\(L(1)\) \(\approx\) \(0.5859592122\)
\(L(\frac12)\) \(\approx\) \(0.5859592122\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 2T^{2} \)
3 \( 1 + 2.87T + 3T^{2} \)
7 \( 1 + 0.347T + 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 - 2.57T + 13T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
23 \( 1 - 2.69T + 23T^{2} \)
29 \( 1 - 6.87T + 29T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 - 4.94T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + 3.90T + 43T^{2} \)
47 \( 1 - 7.29T + 47T^{2} \)
53 \( 1 + 2.83T + 53T^{2} \)
59 \( 1 - 6.30T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 - 7.67T + 67T^{2} \)
71 \( 1 - 9.30T + 71T^{2} \)
73 \( 1 + 1.38T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 14.8T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 9.45T + 97T^{2} \)
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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

−7.925520007989315814113230675667, −6.82183911011556705288425729536, −6.57723296998921608867352947548, −5.70286344696898770416583917933, −4.96010219377992190620177576792, −4.59294923691610902385761941072, −3.59219918406655310276706902359, −2.31980406943432815231010789100, −1.08620044634322925894159128700, −0.60325252527394366487365240037, 0.60325252527394366487365240037, 1.08620044634322925894159128700, 2.31980406943432815231010789100, 3.59219918406655310276706902359, 4.59294923691610902385761941072, 4.96010219377992190620177576792, 5.70286344696898770416583917933, 6.57723296998921608867352947548, 6.82183911011556705288425729536, 7.925520007989315814113230675667

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