Properties

Label 2-10e4-1.1-c1-0-104
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  + 3.09·3-s + 0.0237·7-s + 6.56·9-s − 3.58·11-s − 3.77·13-s + 3.62·17-s + 2.43·19-s + 0.0735·21-s + 1.71·23-s + 11.0·27-s + 3.85·29-s + 6.00·31-s − 11.0·33-s + 0.369·37-s − 11.6·39-s − 7.80·41-s + 0.174·43-s + 7.81·47-s − 6.99·49-s + 11.2·51-s + 8.97·53-s + 7.51·57-s + 4.45·59-s + 9.21·61-s + 0.156·63-s − 4.47·67-s + 5.30·69-s + ⋯
L(s)  = 1  + 1.78·3-s + 0.00899·7-s + 2.18·9-s − 1.08·11-s − 1.04·13-s + 0.878·17-s + 0.557·19-s + 0.0160·21-s + 0.357·23-s + 2.12·27-s + 0.716·29-s + 1.07·31-s − 1.92·33-s + 0.0607·37-s − 1.87·39-s − 1.21·41-s + 0.0266·43-s + 1.13·47-s − 0.999·49-s + 1.56·51-s + 1.23·53-s + 0.995·57-s + 0.580·59-s + 1.17·61-s + 0.0196·63-s − 0.546·67-s + 0.638·69-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

\(L(1)\) \(\approx\) \(4.242773319\)
\(L(\frac12)\) \(\approx\) \(4.242773319\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.09T + 3T^{2} \)
7 \( 1 - 0.0237T + 7T^{2} \)
11 \( 1 + 3.58T + 11T^{2} \)
13 \( 1 + 3.77T + 13T^{2} \)
17 \( 1 - 3.62T + 17T^{2} \)
19 \( 1 - 2.43T + 19T^{2} \)
23 \( 1 - 1.71T + 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 6.00T + 31T^{2} \)
37 \( 1 - 0.369T + 37T^{2} \)
41 \( 1 + 7.80T + 41T^{2} \)
43 \( 1 - 0.174T + 43T^{2} \)
47 \( 1 - 7.81T + 47T^{2} \)
53 \( 1 - 8.97T + 53T^{2} \)
59 \( 1 - 4.45T + 59T^{2} \)
61 \( 1 - 9.21T + 61T^{2} \)
67 \( 1 + 4.47T + 67T^{2} \)
71 \( 1 - 9.69T + 71T^{2} \)
73 \( 1 - 3.95T + 73T^{2} \)
79 \( 1 - 9.68T + 79T^{2} \)
83 \( 1 + 8.95T + 83T^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 + 2.76T + 97T^{2} \)
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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

−7.66765027014601616396410048308, −7.33583725121586646066292947661, −6.56407681461299597351327118533, −5.36545890396885257654235850206, −4.87219848324955404231764080822, −3.98319984817860364240738423251, −3.17056675513176355335272377249, −2.69475242097452430260583585026, −2.05147821787418019361099686621, −0.894786437594806613657316792975, 0.894786437594806613657316792975, 2.05147821787418019361099686621, 2.69475242097452430260583585026, 3.17056675513176355335272377249, 3.98319984817860364240738423251, 4.87219848324955404231764080822, 5.36545890396885257654235850206, 6.56407681461299597351327118533, 7.33583725121586646066292947661, 7.66765027014601616396410048308

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