Properties

Label 2-10e4-1.1-c1-0-195
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  − 0.602·3-s + 3.10·7-s − 2.63·9-s + 6.35·11-s + 1.18·13-s − 8.12·17-s + 3.07·19-s − 1.87·21-s − 4.85·23-s + 3.39·27-s + 1.77·29-s − 6.90·31-s − 3.82·33-s + 6.84·37-s − 0.715·39-s − 5.93·41-s − 4.11·43-s − 1.38·47-s + 2.65·49-s + 4.89·51-s − 10.8·53-s − 1.85·57-s − 10.3·59-s + 8.21·61-s − 8.19·63-s + 0.367·67-s + 2.92·69-s + ⋯
L(s)  = 1  − 0.347·3-s + 1.17·7-s − 0.879·9-s + 1.91·11-s + 0.329·13-s − 1.97·17-s + 0.706·19-s − 0.408·21-s − 1.01·23-s + 0.653·27-s + 0.329·29-s − 1.24·31-s − 0.666·33-s + 1.12·37-s − 0.114·39-s − 0.926·41-s − 0.626·43-s − 0.201·47-s + 0.378·49-s + 0.685·51-s − 1.49·53-s − 0.245·57-s − 1.34·59-s + 1.05·61-s − 1.03·63-s + 0.0449·67-s + 0.352·69-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 \( 1 + 0.602T + 3T^{2} \)
7 \( 1 - 3.10T + 7T^{2} \)
11 \( 1 - 6.35T + 11T^{2} \)
13 \( 1 - 1.18T + 13T^{2} \)
17 \( 1 + 8.12T + 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 - 1.77T + 29T^{2} \)
31 \( 1 + 6.90T + 31T^{2} \)
37 \( 1 - 6.84T + 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 + 4.11T + 43T^{2} \)
47 \( 1 + 1.38T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 8.21T + 61T^{2} \)
67 \( 1 - 0.367T + 67T^{2} \)
71 \( 1 + 7.47T + 71T^{2} \)
73 \( 1 + 5.05T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 3.71T + 83T^{2} \)
89 \( 1 - 8.32T + 89T^{2} \)
97 \( 1 + 4.97T + 97T^{2} \)
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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

−7.22385319688266099393545117044, −6.50139749726416113723075927782, −6.09033577278858618796203328331, −5.24338053023999988005293450768, −4.48697349406640405753153045976, −4.00941910776998967335661511569, −3.03714727809173787816583389908, −1.91931303874710532698478238552, −1.36251584957084959594039619554, 0, 1.36251584957084959594039619554, 1.91931303874710532698478238552, 3.03714727809173787816583389908, 4.00941910776998967335661511569, 4.48697349406640405753153045976, 5.24338053023999988005293450768, 6.09033577278858618796203328331, 6.50139749726416113723075927782, 7.22385319688266099393545117044

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