Properties

Label 2-10e4-1.1-c1-0-160
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  − 0.208·3-s + 0.439·7-s − 2.95·9-s − 4.80·11-s + 3.86·13-s + 3.88·17-s − 0.419·19-s − 0.0913·21-s + 6.40·23-s + 1.23·27-s − 2.19·29-s − 3.99·31-s + 0.999·33-s + 1.63·37-s − 0.803·39-s + 5.36·41-s − 5.14·43-s − 5.02·47-s − 6.80·49-s − 0.807·51-s − 10.1·53-s + 0.0873·57-s − 8.64·59-s − 2.95·61-s − 1.29·63-s + 4.73·67-s − 1.33·69-s + ⋯
L(s)  = 1  − 0.120·3-s + 0.165·7-s − 0.985·9-s − 1.44·11-s + 1.07·13-s + 0.941·17-s − 0.0962·19-s − 0.0199·21-s + 1.33·23-s + 0.238·27-s − 0.407·29-s − 0.718·31-s + 0.173·33-s + 0.268·37-s − 0.128·39-s + 0.837·41-s − 0.784·43-s − 0.732·47-s − 0.972·49-s − 0.113·51-s − 1.39·53-s + 0.0115·57-s − 1.12·59-s − 0.377·61-s − 0.163·63-s + 0.578·67-s − 0.160·69-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 \( 1 + 0.208T + 3T^{2} \)
7 \( 1 - 0.439T + 7T^{2} \)
11 \( 1 + 4.80T + 11T^{2} \)
13 \( 1 - 3.86T + 13T^{2} \)
17 \( 1 - 3.88T + 17T^{2} \)
19 \( 1 + 0.419T + 19T^{2} \)
23 \( 1 - 6.40T + 23T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 + 3.99T + 31T^{2} \)
37 \( 1 - 1.63T + 37T^{2} \)
41 \( 1 - 5.36T + 41T^{2} \)
43 \( 1 + 5.14T + 43T^{2} \)
47 \( 1 + 5.02T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 8.64T + 59T^{2} \)
61 \( 1 + 2.95T + 61T^{2} \)
67 \( 1 - 4.73T + 67T^{2} \)
71 \( 1 + 6.24T + 71T^{2} \)
73 \( 1 - 8.71T + 73T^{2} \)
79 \( 1 - 7.96T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + 9.95T + 97T^{2} \)
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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

−7.60559587825017882831040296711, −6.43997175833278738145118343724, −5.98929615546496200122105535338, −5.14131890412637457576323381095, −4.88964248350768753158873113829, −3.50585685019732830867759958500, −3.16021518666387423959196705221, −2.22808898937280711624430553913, −1.14804446735543912384388682093, 0, 1.14804446735543912384388682093, 2.22808898937280711624430553913, 3.16021518666387423959196705221, 3.50585685019732830867759958500, 4.88964248350768753158873113829, 5.14131890412637457576323381095, 5.98929615546496200122105535338, 6.43997175833278738145118343724, 7.60559587825017882831040296711

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