Properties

Label 2-95e2-1.1-c1-0-338
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  + 2.13·2-s + 2.23·3-s + 2.54·4-s + 4.75·6-s − 1.48·7-s + 1.16·8-s + 1.97·9-s + 4.68·11-s + 5.68·12-s − 0.361·13-s − 3.16·14-s − 2.60·16-s + 5.47·17-s + 4.21·18-s − 3.31·21-s + 9.98·22-s + 6.16·23-s + 2.60·24-s − 0.769·26-s − 2.28·27-s − 3.78·28-s + 0.895·29-s − 4.80·31-s − 7.89·32-s + 10.4·33-s + 11.6·34-s + 5.02·36-s + ⋯
L(s)  = 1  + 1.50·2-s + 1.28·3-s + 1.27·4-s + 1.94·6-s − 0.561·7-s + 0.412·8-s + 0.658·9-s + 1.41·11-s + 1.63·12-s − 0.100·13-s − 0.846·14-s − 0.651·16-s + 1.32·17-s + 0.992·18-s − 0.723·21-s + 2.12·22-s + 1.28·23-s + 0.531·24-s − 0.150·26-s − 0.440·27-s − 0.715·28-s + 0.166·29-s − 0.862·31-s − 1.39·32-s + 1.81·33-s + 2.00·34-s + 0.838·36-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

\(L(1)\) \(\approx\) \(8.424515602\)
\(L(\frac12)\) \(\approx\) \(8.424515602\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.13T + 2T^{2} \)
3 \( 1 - 2.23T + 3T^{2} \)
7 \( 1 + 1.48T + 7T^{2} \)
11 \( 1 - 4.68T + 11T^{2} \)
13 \( 1 + 0.361T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
23 \( 1 - 6.16T + 23T^{2} \)
29 \( 1 - 0.895T + 29T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + 6.52T + 53T^{2} \)
59 \( 1 - 9.80T + 59T^{2} \)
61 \( 1 - 2.34T + 61T^{2} \)
67 \( 1 - 8.85T + 67T^{2} \)
71 \( 1 + 6.41T + 71T^{2} \)
73 \( 1 + 2.86T + 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 + 6.16T + 83T^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 - 4.71T + 97T^{2} \)
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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

−7.51723150280944624346215433470, −6.99506272389604494319247488628, −6.21508109680371900657605140720, −5.66887087839387715422570052084, −4.77304536633881588656759909386, −3.99187835630352805437099911388, −3.48569726050423765640089102074, −2.99555963687108821725985108102, −2.23566997070555733694348710518, −1.10791784860229896630846604771, 1.10791784860229896630846604771, 2.23566997070555733694348710518, 2.99555963687108821725985108102, 3.48569726050423765640089102074, 3.99187835630352805437099911388, 4.77304536633881588656759909386, 5.66887087839387715422570052084, 6.21508109680371900657605140720, 6.99506272389604494319247488628, 7.51723150280944624346215433470

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