Properties

Label 2-70e2-1.1-c1-0-47
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  − 0.364·3-s − 2.86·9-s + 1.36·11-s − 2.63·13-s + 2.23·17-s − 6.23·19-s + 6.59·23-s + 2.13·27-s + 5.50·29-s + 4.50·31-s − 0.497·33-s − 2.09·37-s + 0.960·39-s + 9.32·41-s − 1.86·43-s − 6.86·47-s − 0.813·51-s − 10.1·53-s + 2.27·57-s − 1.63·59-s + 0.0394·61-s − 6.72·67-s − 2.40·69-s − 6.27·71-s − 4·73-s + 5.32·79-s + 7.82·81-s + ⋯
L(s)  = 1  − 0.210·3-s − 0.955·9-s + 0.411·11-s − 0.730·13-s + 0.541·17-s − 1.42·19-s + 1.37·23-s + 0.411·27-s + 1.02·29-s + 0.808·31-s − 0.0865·33-s − 0.344·37-s + 0.153·39-s + 1.45·41-s − 0.284·43-s − 1.00·47-s − 0.113·51-s − 1.39·53-s + 0.300·57-s − 0.212·59-s + 0.00505·61-s − 0.822·67-s − 0.289·69-s − 0.744·71-s − 0.468·73-s + 0.599·79-s + 0.869·81-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: Trivial
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 \( 1 + 0.364T + 3T^{2} \)
11 \( 1 - 1.36T + 11T^{2} \)
13 \( 1 + 2.63T + 13T^{2} \)
17 \( 1 - 2.23T + 17T^{2} \)
19 \( 1 + 6.23T + 19T^{2} \)
23 \( 1 - 6.59T + 23T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 - 4.50T + 31T^{2} \)
37 \( 1 + 2.09T + 37T^{2} \)
41 \( 1 - 9.32T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 1.63T + 59T^{2} \)
61 \( 1 - 0.0394T + 61T^{2} \)
67 \( 1 + 6.72T + 67T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 5.32T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 - 0.867T + 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−8.046164301569252230331340118529, −7.06486407701617631813353106427, −6.45015244278290398413794175758, −5.77819942504301417483198922803, −4.90390435291548281691570260909, −4.32770196285317151248163778856, −3.12780180591418807188250946158, −2.57447042441233340321432350054, −1.28838503923029242039403241500, 0, 1.28838503923029242039403241500, 2.57447042441233340321432350054, 3.12780180591418807188250946158, 4.32770196285317151248163778856, 4.90390435291548281691570260909, 5.77819942504301417483198922803, 6.45015244278290398413794175758, 7.06486407701617631813353106427, 8.046164301569252230331340118529

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