Properties

Label 2-95e2-1.1-c1-0-316
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.906·2-s − 3.21·3-s − 1.17·4-s − 2.91·6-s + 2.59·7-s − 2.88·8-s + 7.35·9-s + 0.741·11-s + 3.78·12-s + 3.78·13-s + 2.35·14-s − 0.258·16-s − 3.16·17-s + 6.67·18-s − 8.35·21-s + 0.672·22-s + 0.570·23-s + 9.27·24-s + 3.43·26-s − 14.0·27-s − 3.05·28-s − 6·29-s − 5.83·31-s + 5.52·32-s − 2.38·33-s − 2.87·34-s − 8.65·36-s + ⋯
L(s)  = 1  + 0.641·2-s − 1.85·3-s − 0.588·4-s − 1.19·6-s + 0.981·7-s − 1.01·8-s + 2.45·9-s + 0.223·11-s + 1.09·12-s + 1.05·13-s + 0.629·14-s − 0.0647·16-s − 0.768·17-s + 1.57·18-s − 1.82·21-s + 0.143·22-s + 0.119·23-s + 1.89·24-s + 0.673·26-s − 2.69·27-s − 0.577·28-s − 1.11·29-s − 1.04·31-s + 0.977·32-s − 0.415·33-s − 0.492·34-s − 1.44·36-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: $\chi_{9025} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.906T + 2T^{2} \)
3 \( 1 + 3.21T + 3T^{2} \)
7 \( 1 - 2.59T + 7T^{2} \)
11 \( 1 - 0.741T + 11T^{2} \)
13 \( 1 - 3.78T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
23 \( 1 - 0.570T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 - 1.40T + 37T^{2} \)
41 \( 1 - 3.83T + 41T^{2} \)
43 \( 1 + 2.59T + 43T^{2} \)
47 \( 1 - 5.08T + 47T^{2} \)
53 \( 1 - 0.160T + 53T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 + 8.57T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 + 4.19T + 83T^{2} \)
89 \( 1 - 16.9T + 89T^{2} \)
97 \( 1 - 3.78T + 97T^{2} \)
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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

−7.18992872350356445315601102949, −6.33337415853949830261093459963, −5.88944113071593742634638173863, −5.32570758629381060427545239496, −4.68467110339083428162079017216, −4.20887088825725285190421617303, −3.49354331073470483610783517566, −1.91699508230189307836811504340, −1.05233988037200921314418825223, 0, 1.05233988037200921314418825223, 1.91699508230189307836811504340, 3.49354331073470483610783517566, 4.20887088825725285190421617303, 4.68467110339083428162079017216, 5.32570758629381060427545239496, 5.88944113071593742634638173863, 6.33337415853949830261093459963, 7.18992872350356445315601102949

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