Properties

Label 2-70e2-1.1-c1-0-63
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.56·3-s + 3.59·9-s − 1.56·11-s − 5.56·13-s − 7.16·17-s + 3.16·19-s − 5.73·23-s + 1.53·27-s + 1.96·29-s + 0.969·31-s − 4.03·33-s + 6.70·37-s − 14.3·39-s − 8.87·41-s + 4.59·43-s − 0.401·47-s − 18.4·51-s − 9.53·53-s + 8.13·57-s − 4.56·59-s + 15.3·61-s − 0.862·67-s − 14.7·69-s − 12.1·71-s − 4·73-s − 12.8·79-s − 6.84·81-s + ⋯
L(s)  = 1  + 1.48·3-s + 1.19·9-s − 0.473·11-s − 1.54·13-s − 1.73·17-s + 0.726·19-s − 1.19·23-s + 0.296·27-s + 0.365·29-s + 0.174·31-s − 0.701·33-s + 1.10·37-s − 2.29·39-s − 1.38·41-s + 0.701·43-s − 0.0584·47-s − 2.57·51-s − 1.31·53-s + 1.07·57-s − 0.594·59-s + 1.95·61-s − 0.105·67-s − 1.77·69-s − 1.44·71-s − 0.468·73-s − 1.44·79-s − 0.760·81-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: \(1\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 + 5.56T + 13T^{2} \)
17 \( 1 + 7.16T + 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 - 1.96T + 29T^{2} \)
31 \( 1 - 0.969T + 31T^{2} \)
37 \( 1 - 6.70T + 37T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 - 4.59T + 43T^{2} \)
47 \( 1 + 0.401T + 47T^{2} \)
53 \( 1 + 9.53T + 53T^{2} \)
59 \( 1 + 4.56T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 0.862T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 5.59T + 89T^{2} \)
97 \( 1 + 0.233T + 97T^{2} \)
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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

−7.958647562123971559295581853445, −7.37846148349129558650874429613, −6.72621762316305402496610937318, −5.68518648548065216559944735124, −4.66807940086336215300358321767, −4.17507797170663660134930533199, −3.04021801066998193085758858297, −2.52109966762684419845790942782, −1.79735645751241287585330547448, 0, 1.79735645751241287585330547448, 2.52109966762684419845790942782, 3.04021801066998193085758858297, 4.17507797170663660134930533199, 4.66807940086336215300358321767, 5.68518648548065216559944735124, 6.72621762316305402496610937318, 7.37846148349129558650874429613, 7.958647562123971559295581853445

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