Properties

Label 2-10e4-1.1-c1-0-202
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·3-s − 3.59·7-s + 2.30·9-s + 0.497·11-s + 2.64·13-s + 5.10·17-s + 0.987·19-s − 8.27·21-s − 6.41·23-s − 1.61·27-s − 5.57·29-s − 6.05·31-s + 1.14·33-s − 4.59·37-s + 6.09·39-s − 2.87·41-s + 9.48·43-s − 5.36·47-s + 5.91·49-s + 11.7·51-s − 0.307·53-s + 2.27·57-s + 1.26·59-s − 6.22·61-s − 8.26·63-s + 5.28·67-s − 14.7·69-s + ⋯
L(s)  = 1  + 1.32·3-s − 1.35·7-s + 0.766·9-s + 0.150·11-s + 0.734·13-s + 1.23·17-s + 0.226·19-s − 1.80·21-s − 1.33·23-s − 0.309·27-s − 1.03·29-s − 1.08·31-s + 0.199·33-s − 0.755·37-s + 0.976·39-s − 0.448·41-s + 1.44·43-s − 0.783·47-s + 0.845·49-s + 1.64·51-s − 0.0422·53-s + 0.301·57-s + 0.164·59-s − 0.797·61-s − 1.04·63-s + 0.645·67-s − 1.77·69-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: \(1\)
Selberg data: \((2,\ 10000,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good3 \( 1 - 2.30T + 3T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 - 0.497T + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 - 0.987T + 19T^{2} \)
23 \( 1 + 6.41T + 23T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 + 4.59T + 37T^{2} \)
41 \( 1 + 2.87T + 41T^{2} \)
43 \( 1 - 9.48T + 43T^{2} \)
47 \( 1 + 5.36T + 47T^{2} \)
53 \( 1 + 0.307T + 53T^{2} \)
59 \( 1 - 1.26T + 59T^{2} \)
61 \( 1 + 6.22T + 61T^{2} \)
67 \( 1 - 5.28T + 67T^{2} \)
71 \( 1 - 0.151T + 71T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 0.849T + 97T^{2} \)
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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

−7.53477147672866759966361898009, −6.72070884237846893776316919582, −5.94840815987068846423057990900, −5.47199250375060653777718638589, −4.12406167649503855639469709753, −3.53665147352019547926548607936, −3.23980808879627899324121257472, −2.29873359018530774252567518132, −1.43574394158407162957834559740, 0, 1.43574394158407162957834559740, 2.29873359018530774252567518132, 3.23980808879627899324121257472, 3.53665147352019547926548607936, 4.12406167649503855639469709753, 5.47199250375060653777718638589, 5.94840815987068846423057990900, 6.72070884237846893776316919582, 7.53477147672866759966361898009

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