Properties

Label 2-75e2-1.1-c1-0-64
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-s − 4-s + 4.47·7-s − 3·8-s + 1.23·11-s − 5.61·13-s + 4.47·14-s − 16-s + 3.85·17-s + 1.23·19-s + 1.23·22-s + 4.47·23-s − 5.61·26-s − 4.47·28-s − 6.61·29-s + 2.76·31-s + 5·32-s + 3.85·34-s + 3.09·37-s + 1.23·38-s + 3.61·41-s − 7.70·43-s − 1.23·44-s + 4.47·46-s − 0.763·47-s + 13.0·49-s + 5.61·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s + 1.69·7-s − 1.06·8-s + 0.372·11-s − 1.55·13-s + 1.19·14-s − 0.250·16-s + 0.934·17-s + 0.283·19-s + 0.263·22-s + 0.932·23-s − 1.10·26-s − 0.845·28-s − 1.22·29-s + 0.496·31-s + 0.883·32-s + 0.660·34-s + 0.508·37-s + 0.200·38-s + 0.565·41-s − 1.17·43-s − 0.186·44-s + 0.659·46-s − 0.111·47-s + 1.85·49-s + 0.779·52-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

\(L(1)\) \(\approx\) \(2.774878795\)
\(L(\frac12)\) \(\approx\) \(2.774878795\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 2T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + 5.61T + 13T^{2} \)
17 \( 1 - 3.85T + 17T^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 6.61T + 29T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
41 \( 1 - 3.61T + 41T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 + 0.763T + 47T^{2} \)
53 \( 1 - 3.61T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 1.61T + 61T^{2} \)
67 \( 1 + 0.763T + 67T^{2} \)
71 \( 1 - 5.23T + 71T^{2} \)
73 \( 1 - 8.09T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + 5.38T + 89T^{2} \)
97 \( 1 + 2.14T + 97T^{2} \)
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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

−8.001792396772810485823347346665, −7.55980602276251211322644190509, −6.70137965771809788979893282411, −5.54358709426977098850322106047, −5.18295119847733815065438716795, −4.62885844592880637460708919260, −3.88282932023147340642541278734, −2.92712935824126354231928550165, −1.96273664905126820344905425208, −0.823830784869458668129635927313, 0.823830784869458668129635927313, 1.96273664905126820344905425208, 2.92712935824126354231928550165, 3.88282932023147340642541278734, 4.62885844592880637460708919260, 5.18295119847733815065438716795, 5.54358709426977098850322106047, 6.70137965771809788979893282411, 7.55980602276251211322644190509, 8.001792396772810485823347346665

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