Properties

Label 2-75e2-1.1-c1-0-5
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  − 2.01·2-s + 2.07·4-s − 1.01·7-s − 0.153·8-s − 4.75·11-s + 0.103·13-s + 2.05·14-s − 3.84·16-s − 5.83·17-s − 0.724·19-s + 9.60·22-s − 9.07·23-s − 0.209·26-s − 2.11·28-s + 3.98·29-s + 1.06·31-s + 8.06·32-s + 11.7·34-s − 4.02·37-s + 1.46·38-s − 7.20·41-s − 8.62·43-s − 9.88·44-s + 18.3·46-s − 8.19·47-s − 5.96·49-s + 0.215·52-s + ⋯
L(s)  = 1  − 1.42·2-s + 1.03·4-s − 0.385·7-s − 0.0541·8-s − 1.43·11-s + 0.0287·13-s + 0.549·14-s − 0.960·16-s − 1.41·17-s − 0.166·19-s + 2.04·22-s − 1.89·23-s − 0.0411·26-s − 0.399·28-s + 0.740·29-s + 0.191·31-s + 1.42·32-s + 2.02·34-s − 0.661·37-s + 0.237·38-s − 1.12·41-s − 1.31·43-s − 1.48·44-s + 2.70·46-s − 1.19·47-s − 0.851·49-s + 0.0298·52-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

\(L(1)\) \(\approx\) \(0.2140371521\)
\(L(\frac12)\) \(\approx\) \(0.2140371521\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.01T + 2T^{2} \)
7 \( 1 + 1.01T + 7T^{2} \)
11 \( 1 + 4.75T + 11T^{2} \)
13 \( 1 - 0.103T + 13T^{2} \)
17 \( 1 + 5.83T + 17T^{2} \)
19 \( 1 + 0.724T + 19T^{2} \)
23 \( 1 + 9.07T + 23T^{2} \)
29 \( 1 - 3.98T + 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 + 4.02T + 37T^{2} \)
41 \( 1 + 7.20T + 41T^{2} \)
43 \( 1 + 8.62T + 43T^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + 4.36T + 53T^{2} \)
59 \( 1 - 4.91T + 59T^{2} \)
61 \( 1 - 6.96T + 61T^{2} \)
67 \( 1 - 9.91T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 + 8.63T + 73T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 - 4.24T + 83T^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + 6.69T + 97T^{2} \)
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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

−8.291663089369118723276607919652, −7.75498571426441115952170904939, −6.74008252144289414785470668438, −6.44873035176682306068245891843, −5.24036555061593347705633360635, −4.58198061543401459934321434748, −3.49477456506453883719513429316, −2.39489810737147324306282910965, −1.81642737932227625191335261394, −0.29046410226323060869455092292, 0.29046410226323060869455092292, 1.81642737932227625191335261394, 2.39489810737147324306282910965, 3.49477456506453883719513429316, 4.58198061543401459934321434748, 5.24036555061593347705633360635, 6.44873035176682306068245891843, 6.74008252144289414785470668438, 7.75498571426441115952170904939, 8.291663089369118723276607919652

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