Properties

Label 2-70e2-1.1-c1-0-39
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  + 3·3-s + 6·9-s + 3·11-s + 13-s − 5·17-s + 8·19-s − 2·23-s + 9·27-s − 29-s + 2·31-s + 9·33-s − 10·37-s + 3·39-s + 6·41-s + 4·43-s + 11·47-s − 15·51-s − 6·53-s + 24·57-s + 10·59-s + 10·67-s − 6·69-s − 10·73-s − 7·79-s + 9·81-s + 12·83-s − 3·87-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s + 0.904·11-s + 0.277·13-s − 1.21·17-s + 1.83·19-s − 0.417·23-s + 1.73·27-s − 0.185·29-s + 0.359·31-s + 1.56·33-s − 1.64·37-s + 0.480·39-s + 0.937·41-s + 0.609·43-s + 1.60·47-s − 2.10·51-s − 0.824·53-s + 3.17·57-s + 1.30·59-s + 1.22·67-s − 0.722·69-s − 1.17·73-s − 0.787·79-s + 81-s + 1.31·83-s − 0.321·87-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

\(L(1)\) \(\approx\) \(4.297126691\)
\(L(\frac12)\) \(\approx\) \(4.297126691\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−8.447504145811414138752894965378, −7.52917575815651472519709129717, −7.14216994640797315588149549045, −6.27196909190688157452279416067, −5.22190557404471392361008817335, −4.17348321895597456565198399478, −3.71092698627393738833608367309, −2.87346274040664131988070140534, −2.07911953341898356527538818753, −1.13784823851991294530611763263, 1.13784823851991294530611763263, 2.07911953341898356527538818753, 2.87346274040664131988070140534, 3.71092698627393738833608367309, 4.17348321895597456565198399478, 5.22190557404471392361008817335, 6.27196909190688157452279416067, 7.14216994640797315588149549045, 7.52917575815651472519709129717, 8.447504145811414138752894965378

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