Properties

Label 2-200-1.1-c5-0-23
Degree $2$
Conductor $200$
Sign $-1$
Analytic cond. $32.0767$
Root an. cond. $5.66363$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 24.1·3-s − 179.·7-s + 339.·9-s − 653.·11-s + 284.·13-s − 383.·17-s − 2.56e3·19-s − 4.34e3·21-s − 948.·23-s + 2.33e3·27-s − 1.52e3·29-s + 3.10e3·31-s − 1.57e4·33-s − 9.99e3·37-s + 6.87e3·39-s + 1.51e4·41-s + 1.75e3·43-s + 1.47e4·47-s + 1.55e4·49-s − 9.26e3·51-s − 8.70e3·53-s − 6.18e4·57-s − 1.26e4·59-s + 4.30e4·61-s − 6.10e4·63-s − 2.61e4·67-s − 2.28e4·69-s + ⋯
L(s)  = 1  + 1.54·3-s − 1.38·7-s + 1.39·9-s − 1.62·11-s + 0.467·13-s − 0.321·17-s − 1.62·19-s − 2.14·21-s − 0.373·23-s + 0.615·27-s − 0.336·29-s + 0.580·31-s − 2.52·33-s − 1.19·37-s + 0.723·39-s + 1.40·41-s + 0.144·43-s + 0.974·47-s + 0.925·49-s − 0.498·51-s − 0.425·53-s − 2.52·57-s − 0.472·59-s + 1.48·61-s − 1.93·63-s − 0.711·67-s − 0.578·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(32.0767\)
Root analytic conductor: \(5.66363\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 200,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 24.1T + 243T^{2} \)
7 \( 1 + 179.T + 1.68e4T^{2} \)
11 \( 1 + 653.T + 1.61e5T^{2} \)
13 \( 1 - 284.T + 3.71e5T^{2} \)
17 \( 1 + 383.T + 1.41e6T^{2} \)
19 \( 1 + 2.56e3T + 2.47e6T^{2} \)
23 \( 1 + 948.T + 6.43e6T^{2} \)
29 \( 1 + 1.52e3T + 2.05e7T^{2} \)
31 \( 1 - 3.10e3T + 2.86e7T^{2} \)
37 \( 1 + 9.99e3T + 6.93e7T^{2} \)
41 \( 1 - 1.51e4T + 1.15e8T^{2} \)
43 \( 1 - 1.75e3T + 1.47e8T^{2} \)
47 \( 1 - 1.47e4T + 2.29e8T^{2} \)
53 \( 1 + 8.70e3T + 4.18e8T^{2} \)
59 \( 1 + 1.26e4T + 7.14e8T^{2} \)
61 \( 1 - 4.30e4T + 8.44e8T^{2} \)
67 \( 1 + 2.61e4T + 1.35e9T^{2} \)
71 \( 1 + 4.62e4T + 1.80e9T^{2} \)
73 \( 1 + 5.13e4T + 2.07e9T^{2} \)
79 \( 1 - 3.93e4T + 3.07e9T^{2} \)
83 \( 1 + 6.95e4T + 3.93e9T^{2} \)
89 \( 1 + 1.30e4T + 5.58e9T^{2} \)
97 \( 1 + 2.62e4T + 8.58e9T^{2} \)
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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

−10.72683375626747405376532993404, −9.980733139001486373688611927880, −8.964095800241609125503822084638, −8.232452920220068416707298089912, −7.18470605974483117427763493836, −5.95708178611450055843294101208, −4.18167679553354898658580796234, −3.05725161085548370452052666276, −2.24820550445289668661267188311, 0, 2.24820550445289668661267188311, 3.05725161085548370452052666276, 4.18167679553354898658580796234, 5.95708178611450055843294101208, 7.18470605974483117427763493836, 8.232452920220068416707298089912, 8.964095800241609125503822084638, 9.980733139001486373688611927880, 10.72683375626747405376532993404

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