Properties

Label 2-10e4-1.1-c1-0-164
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  + 1.67·3-s + 3.57·7-s − 0.177·9-s + 2.43·11-s + 4.65·13-s + 6.59·17-s + 3.50·19-s + 6.00·21-s + 4.36·23-s − 5.33·27-s + 3.64·29-s + 2.93·31-s + 4.08·33-s − 5.33·37-s + 7.82·39-s − 0.295·41-s − 6.57·43-s − 7.34·47-s + 5.79·49-s + 11.0·51-s − 0.486·53-s + 5.88·57-s + 8.80·59-s + 9.00·61-s − 0.636·63-s − 10.0·67-s + 7.33·69-s + ⋯
L(s)  = 1  + 0.969·3-s + 1.35·7-s − 0.0593·9-s + 0.732·11-s + 1.29·13-s + 1.59·17-s + 0.804·19-s + 1.31·21-s + 0.910·23-s − 1.02·27-s + 0.677·29-s + 0.526·31-s + 0.710·33-s − 0.876·37-s + 1.25·39-s − 0.0462·41-s − 1.00·43-s − 1.07·47-s + 0.828·49-s + 1.55·51-s − 0.0668·53-s + 0.779·57-s + 1.14·59-s + 1.15·61-s − 0.0802·63-s − 1.22·67-s + 0.883·69-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

\(L(1)\) \(\approx\) \(4.643229788\)
\(L(\frac12)\) \(\approx\) \(4.643229788\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.67T + 3T^{2} \)
7 \( 1 - 3.57T + 7T^{2} \)
11 \( 1 - 2.43T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 - 6.59T + 17T^{2} \)
19 \( 1 - 3.50T + 19T^{2} \)
23 \( 1 - 4.36T + 23T^{2} \)
29 \( 1 - 3.64T + 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 + 5.33T + 37T^{2} \)
41 \( 1 + 0.295T + 41T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 + 0.486T + 53T^{2} \)
59 \( 1 - 8.80T + 59T^{2} \)
61 \( 1 - 9.00T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 7.53T + 79T^{2} \)
83 \( 1 - 1.56T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.892980912920484005933767644307, −7.13513437844808925482330048280, −6.35292293308696356026593170158, −5.43376284786903114541579400193, −5.01565354390551302508907590772, −3.93017431620062842154885738604, −3.42872220764277888711019464267, −2.71300452347304465814777103448, −1.52044654261762649398528719080, −1.16501712302622099934802015926, 1.16501712302622099934802015926, 1.52044654261762649398528719080, 2.71300452347304465814777103448, 3.42872220764277888711019464267, 3.93017431620062842154885738604, 5.01565354390551302508907590772, 5.43376284786903114541579400193, 6.35292293308696356026593170158, 7.13513437844808925482330048280, 7.892980912920484005933767644307

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