Properties

Label 2-75e2-1.1-c1-0-80
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37·2-s + 3.64·4-s − 4.26·7-s − 3.91·8-s − 2.49·11-s − 2.02·13-s + 10.1·14-s + 2.01·16-s + 7.24·17-s + 3.26·19-s + 5.92·22-s − 6.15·23-s + 4.82·26-s − 15.5·28-s + 0.951·29-s + 2.66·31-s + 3.05·32-s − 17.2·34-s − 9.66·37-s − 7.76·38-s + 12.1·41-s − 7.95·43-s − 9.08·44-s + 14.6·46-s − 2.93·47-s + 11.1·49-s − 7.40·52-s + ⋯
L(s)  = 1  − 1.68·2-s + 1.82·4-s − 1.61·7-s − 1.38·8-s − 0.751·11-s − 0.562·13-s + 2.70·14-s + 0.502·16-s + 1.75·17-s + 0.749·19-s + 1.26·22-s − 1.28·23-s + 0.946·26-s − 2.94·28-s + 0.176·29-s + 0.478·31-s + 0.539·32-s − 2.95·34-s − 1.58·37-s − 1.25·38-s + 1.90·41-s − 1.21·43-s − 1.37·44-s + 2.15·46-s − 0.428·47-s + 1.59·49-s − 1.02·52-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: $\chi_{5625} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 2.37T + 2T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 + 2.49T + 11T^{2} \)
13 \( 1 + 2.02T + 13T^{2} \)
17 \( 1 - 7.24T + 17T^{2} \)
19 \( 1 - 3.26T + 19T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
29 \( 1 - 0.951T + 29T^{2} \)
31 \( 1 - 2.66T + 31T^{2} \)
37 \( 1 + 9.66T + 37T^{2} \)
41 \( 1 - 12.1T + 41T^{2} \)
43 \( 1 + 7.95T + 43T^{2} \)
47 \( 1 + 2.93T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 8.13T + 59T^{2} \)
61 \( 1 + 3.33T + 61T^{2} \)
67 \( 1 - 11.2T + 67T^{2} \)
71 \( 1 - 9.67T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 0.997T + 89T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−7.79651808839676823901564035182, −7.37505908861705994575040316237, −6.60153821390089187840978340831, −5.91988960049463262049751744978, −5.13018409897203650551329339766, −3.69782511540727701415131968362, −2.99884104090010664696669841388, −2.19637742640192218530850196446, −0.936072746479354619713873049443, 0, 0.936072746479354619713873049443, 2.19637742640192218530850196446, 2.99884104090010664696669841388, 3.69782511540727701415131968362, 5.13018409897203650551329339766, 5.91988960049463262049751744978, 6.60153821390089187840978340831, 7.37505908861705994575040316237, 7.79651808839676823901564035182

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