Properties

Label 2-75e2-1.1-c1-0-4
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 − 3.23·11-s − 3.38·13-s − 4.47·14-s − 16-s − 2.85·17-s − 3.23·19-s − 3.23·22-s − 4.47·23-s − 3.38·26-s + 4.47·28-s − 4.38·29-s + 7.23·31-s + 5·32-s − 2.85·34-s − 8.09·37-s − 3.23·38-s + 1.38·41-s + 5.70·43-s + 3.23·44-s − 4.47·46-s − 5.23·47-s + 13.0·49-s + 3.38·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s − 1.69·7-s − 1.06·8-s − 0.975·11-s − 0.937·13-s − 1.19·14-s − 0.250·16-s − 0.692·17-s − 0.742·19-s − 0.689·22-s − 0.932·23-s − 0.663·26-s + 0.845·28-s − 0.813·29-s + 1.29·31-s + 0.883·32-s − 0.489·34-s − 1.33·37-s − 0.524·38-s + 0.215·41-s + 0.870·43-s + 0.487·44-s − 0.659·46-s − 0.763·47-s + 1.85·49-s + 0.468·52-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

\(L(1)\) \(\approx\) \(0.3565347511\)
\(L(\frac12)\) \(\approx\) \(0.3565347511\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + 3.38T + 13T^{2} \)
17 \( 1 + 2.85T + 17T^{2} \)
19 \( 1 + 3.23T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 - 7.23T + 31T^{2} \)
37 \( 1 + 8.09T + 37T^{2} \)
41 \( 1 - 1.38T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + 5.23T + 47T^{2} \)
53 \( 1 - 1.38T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 0.618T + 61T^{2} \)
67 \( 1 + 5.23T + 67T^{2} \)
71 \( 1 - 0.763T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + 7.61T + 89T^{2} \)
97 \( 1 + 8.85T + 97T^{2} \)
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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

−8.176919069173932578258248988207, −7.27287013267403711418250434136, −6.51175060940829629699506906174, −5.96223337413160266656424768806, −5.22117213363993409611126866196, −4.43724843278272267611178047048, −3.74255169596848897079889402937, −2.92014097214513118854070387592, −2.30222143931357878954969133679, −0.26217165973092125805544686250, 0.26217165973092125805544686250, 2.30222143931357878954969133679, 2.92014097214513118854070387592, 3.74255169596848897079889402937, 4.43724843278272267611178047048, 5.22117213363993409611126866196, 5.96223337413160266656424768806, 6.51175060940829629699506906174, 7.27287013267403711418250434136, 8.176919069173932578258248988207

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