Properties

Label 2-10e4-1.1-c1-0-76
Degree $2$
Conductor $10000$
Sign $1$
Analytic cond. $79.8504$
Root an. cond. $8.93590$
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  + 0.710·3-s + 4.59·7-s − 2.49·9-s − 3.91·11-s − 0.572·13-s − 0.232·17-s + 5.55·19-s + 3.26·21-s + 4.93·23-s − 3.90·27-s + 4.13·29-s + 3.49·31-s − 2.78·33-s − 5.41·37-s − 0.406·39-s + 10.4·41-s − 1.38·43-s − 0.920·47-s + 14.0·49-s − 0.165·51-s + 1.23·53-s + 3.94·57-s − 4.50·59-s − 11.6·61-s − 11.4·63-s + 2.95·67-s + 3.50·69-s + ⋯
L(s)  = 1  + 0.410·3-s + 1.73·7-s − 0.831·9-s − 1.18·11-s − 0.158·13-s − 0.0564·17-s + 1.27·19-s + 0.711·21-s + 1.02·23-s − 0.751·27-s + 0.767·29-s + 0.627·31-s − 0.484·33-s − 0.890·37-s − 0.0651·39-s + 1.62·41-s − 0.211·43-s − 0.134·47-s + 2.01·49-s − 0.0231·51-s + 0.169·53-s + 0.522·57-s − 0.586·59-s − 1.49·61-s − 1.44·63-s + 0.360·67-s + 0.421·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10000\)    =    \(2^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(79.8504\)
Root analytic conductor: \(8.93590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10000,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.775123655\)
\(L(\frac12)\) \(\approx\) \(2.775123655\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 - 0.710T + 3T^{2} \)
7 \( 1 - 4.59T + 7T^{2} \)
11 \( 1 + 3.91T + 11T^{2} \)
13 \( 1 + 0.572T + 13T^{2} \)
17 \( 1 + 0.232T + 17T^{2} \)
19 \( 1 - 5.55T + 19T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 + 1.38T + 43T^{2} \)
47 \( 1 + 0.920T + 47T^{2} \)
53 \( 1 - 1.23T + 53T^{2} \)
59 \( 1 + 4.50T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 2.95T + 67T^{2} \)
71 \( 1 + 3.20T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 - 9.61T + 79T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 - 7.25T + 89T^{2} \)
97 \( 1 + 8.31T + 97T^{2} \)
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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

−7.70771788408757860969617906667, −7.33992876858213333193747100270, −6.20034299265654490649503150764, −5.33222881336350041348589742916, −5.05792483021796117913322691554, −4.34186310814213649840383803376, −3.15482459882431132752101276775, −2.67483088668300646057002736339, −1.78582458748539910249917493807, −0.78470304251297588239721788651, 0.78470304251297588239721788651, 1.78582458748539910249917493807, 2.67483088668300646057002736339, 3.15482459882431132752101276775, 4.34186310814213649840383803376, 5.05792483021796117913322691554, 5.33222881336350041348589742916, 6.20034299265654490649503150764, 7.33992876858213333193747100270, 7.70771788408757860969617906667

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