Properties

Label 2-10e4-1.1-c1-0-20
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  − 1.34·3-s − 0.833·7-s − 1.18·9-s − 0.833·11-s − 4.56·13-s − 5.45·17-s + 8.65·19-s + 1.12·21-s + 3.53·23-s + 5.63·27-s + 3.26·29-s − 6.00·31-s + 1.12·33-s − 7.31·37-s + 6.15·39-s + 1.86·41-s − 1.63·43-s − 0.833·47-s − 6.30·49-s + 7.35·51-s − 6.42·53-s − 11.6·57-s − 13.5·59-s + 5.88·61-s + 0.984·63-s + 1.49·67-s − 4.76·69-s + ⋯
L(s)  = 1  − 0.778·3-s − 0.314·7-s − 0.393·9-s − 0.251·11-s − 1.26·13-s − 1.32·17-s + 1.98·19-s + 0.245·21-s + 0.736·23-s + 1.08·27-s + 0.607·29-s − 1.07·31-s + 0.195·33-s − 1.20·37-s + 0.985·39-s + 0.291·41-s − 0.249·43-s − 0.121·47-s − 0.900·49-s + 1.02·51-s − 0.882·53-s − 1.54·57-s − 1.76·59-s + 0.753·61-s + 0.124·63-s + 0.183·67-s − 0.573·69-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

\(L(1)\) \(\approx\) \(0.6271241866\)
\(L(\frac12)\) \(\approx\) \(0.6271241866\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.34T + 3T^{2} \)
7 \( 1 + 0.833T + 7T^{2} \)
11 \( 1 + 0.833T + 11T^{2} \)
13 \( 1 + 4.56T + 13T^{2} \)
17 \( 1 + 5.45T + 17T^{2} \)
19 \( 1 - 8.65T + 19T^{2} \)
23 \( 1 - 3.53T + 23T^{2} \)
29 \( 1 - 3.26T + 29T^{2} \)
31 \( 1 + 6.00T + 31T^{2} \)
37 \( 1 + 7.31T + 37T^{2} \)
41 \( 1 - 1.86T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + 0.833T + 47T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 5.88T + 61T^{2} \)
67 \( 1 - 1.49T + 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 - 3.33T + 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 7.00T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−7.45321260039410978104404576580, −6.94594081939532416366924066712, −6.35413232769941600572470876888, −5.37135069101951160079606405748, −5.16242252188063419691036240488, −4.39917272164869624707321581506, −3.21625065019350257804145380109, −2.75865822108879107173767190997, −1.62567419204546333727487084347, −0.38301194296712549894306772612, 0.38301194296712549894306772612, 1.62567419204546333727487084347, 2.75865822108879107173767190997, 3.21625065019350257804145380109, 4.39917272164869624707321581506, 5.16242252188063419691036240488, 5.37135069101951160079606405748, 6.35413232769941600572470876888, 6.94594081939532416366924066712, 7.45321260039410978104404576580

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