Properties

Label 2-70e2-1.1-c1-0-3
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s − 3·11-s − 2·13-s − 3·17-s − 19-s − 3·23-s + 5·27-s − 6·29-s − 7·31-s + 3·33-s + 37-s + 2·39-s + 6·41-s + 4·43-s + 9·47-s + 3·51-s − 3·53-s + 57-s + 9·59-s − 61-s + 7·67-s + 3·69-s + 73-s − 13·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 0.625·23-s + 0.962·27-s − 1.11·29-s − 1.25·31-s + 0.522·33-s + 0.164·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.31·47-s + 0.420·51-s − 0.412·53-s + 0.132·57-s + 1.17·59-s − 0.128·61-s + 0.855·67-s + 0.361·69-s + 0.117·73-s − 1.46·79-s + 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7009066754\)
\(L(\frac12)\) \(\approx\) \(0.7009066754\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.206314383565699032583413773277, −7.50583850856640237230837436037, −6.85420158442897404526323646241, −5.81685939880161480103979342246, −5.56063789989129311990774760162, −4.65518649907072722746944298509, −3.83503074284329452660005843394, −2.72320977868378818164957546956, −2.04516451501611384471824498227, −0.44464023562677003765543182427, 0.44464023562677003765543182427, 2.04516451501611384471824498227, 2.72320977868378818164957546956, 3.83503074284329452660005843394, 4.65518649907072722746944298509, 5.56063789989129311990774760162, 5.81685939880161480103979342246, 6.85420158442897404526323646241, 7.50583850856640237230837436037, 8.206314383565699032583413773277

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