Properties

Label 2-70e2-1.1-c1-0-23
Degree $2$
Conductor $4900$
Sign $1$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
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  + 1.73·3-s + 2.27·11-s − 6.09·13-s + 4.77·17-s + 4.27·19-s − 0.894·23-s − 5.19·27-s − 3.27·29-s + 4.27·31-s + 3.94·33-s + 5.61·37-s − 10.5·39-s + 11.2·41-s + 6.50·43-s + 2.15·47-s + 8.27·51-s − 7.40·53-s + 7.40·57-s − 4.27·59-s − 1.54·61-s + 13.9·67-s − 1.54·69-s + 10.5·71-s + 2.15·73-s − 0.274·79-s − 9·81-s + 5.67·83-s + ⋯
L(s)  = 1  + 1.00·3-s + 0.685·11-s − 1.68·13-s + 1.15·17-s + 0.980·19-s − 0.186·23-s − 1.00·27-s − 0.608·29-s + 0.767·31-s + 0.685·33-s + 0.923·37-s − 1.68·39-s + 1.76·41-s + 0.992·43-s + 0.313·47-s + 1.15·51-s − 1.01·53-s + 0.980·57-s − 0.556·59-s − 0.198·61-s + 1.69·67-s − 0.186·69-s + 1.25·71-s + 0.251·73-s − 0.0309·79-s − 81-s + 0.622·83-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: no
Arithmetic: yes
Character: $\chi_{4900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.764705749\)
\(L(\frac12)\) \(\approx\) \(2.764705749\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 3T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 - 4.77T + 17T^{2} \)
19 \( 1 - 4.27T + 19T^{2} \)
23 \( 1 + 0.894T + 23T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 - 4.27T + 31T^{2} \)
37 \( 1 - 5.61T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 6.50T + 43T^{2} \)
47 \( 1 - 2.15T + 47T^{2} \)
53 \( 1 + 7.40T + 53T^{2} \)
59 \( 1 + 4.27T + 59T^{2} \)
61 \( 1 + 1.54T + 61T^{2} \)
67 \( 1 - 13.9T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 2.15T + 73T^{2} \)
79 \( 1 + 0.274T + 79T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 - 7T + 89T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−7.967829617730661040734651414324, −7.80321981349920408985286262097, −7.07134674960913081994830673187, −6.04747944773841036638499087368, −5.33678476938110887404819960419, −4.44899657437765326272886649768, −3.59045576764620882413256763925, −2.83297548575529965345047392015, −2.16426371358390404195355808420, −0.873968337279449952461044586681, 0.873968337279449952461044586681, 2.16426371358390404195355808420, 2.83297548575529965345047392015, 3.59045576764620882413256763925, 4.44899657437765326272886649768, 5.33678476938110887404819960419, 6.04747944773841036638499087368, 7.07134674960913081994830673187, 7.80321981349920408985286262097, 7.967829617730661040734651414324

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