Properties

Label 2-75e2-1.1-c1-0-7
Degree $2$
Conductor $5625$
Sign $1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
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  − 0.246·2-s − 1.93·4-s − 1.24·7-s + 0.972·8-s − 2.56·11-s − 4.68·13-s + 0.307·14-s + 3.63·16-s − 5.83·17-s + 4.16·19-s + 0.634·22-s − 1.60·23-s + 1.15·26-s + 2.41·28-s + 3.21·29-s − 9.19·31-s − 2.84·32-s + 1.44·34-s − 1.27·37-s − 1.02·38-s − 8.69·41-s − 3.88·43-s + 4.98·44-s + 0.396·46-s + 3.20·47-s − 5.44·49-s + 9.07·52-s + ⋯
L(s)  = 1  − 0.174·2-s − 0.969·4-s − 0.471·7-s + 0.343·8-s − 0.774·11-s − 1.29·13-s + 0.0822·14-s + 0.909·16-s − 1.41·17-s + 0.956·19-s + 0.135·22-s − 0.334·23-s + 0.226·26-s + 0.456·28-s + 0.597·29-s − 1.65·31-s − 0.502·32-s + 0.247·34-s − 0.210·37-s − 0.166·38-s − 1.35·41-s − 0.591·43-s + 0.751·44-s + 0.0584·46-s + 0.468·47-s − 0.777·49-s + 1.25·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4664405130\)
\(L(\frac12)\) \(\approx\) \(0.4664405130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 0.246T + 2T^{2} \)
7 \( 1 + 1.24T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 + 4.68T + 13T^{2} \)
17 \( 1 + 5.83T + 17T^{2} \)
19 \( 1 - 4.16T + 19T^{2} \)
23 \( 1 + 1.60T + 23T^{2} \)
29 \( 1 - 3.21T + 29T^{2} \)
31 \( 1 + 9.19T + 31T^{2} \)
37 \( 1 + 1.27T + 37T^{2} \)
41 \( 1 + 8.69T + 41T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 - 3.82T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 - 2.83T + 83T^{2} \)
89 \( 1 - 1.38T + 89T^{2} \)
97 \( 1 - 0.0305T + 97T^{2} \)
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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

−8.230871937434648291252062786485, −7.37787945554749536776342719534, −6.96376470095587228264092634269, −5.82310125382323565916426903800, −5.14470687218138336572438520658, −4.61304142820689229425628321706, −3.70627129060984736909146602527, −2.84718463169421640630631421969, −1.87317647987066047457216607159, −0.36046574005908201012212072555, 0.36046574005908201012212072555, 1.87317647987066047457216607159, 2.84718463169421640630631421969, 3.70627129060984736909146602527, 4.61304142820689229425628321706, 5.14470687218138336572438520658, 5.82310125382323565916426903800, 6.96376470095587228264092634269, 7.37787945554749536776342719534, 8.230871937434648291252062786485

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