Properties

Label 2-20e2-1.1-c5-0-26
Degree $2$
Conductor $400$
Sign $-1$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
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  − 19.5·3-s + 35.0·7-s + 138.·9-s + 426.·11-s − 1.10e3·13-s − 109.·17-s − 495.·19-s − 684.·21-s + 2.49e3·23-s + 2.04e3·27-s − 42.4·29-s + 7.99e3·31-s − 8.31e3·33-s + 1.37e4·37-s + 2.15e4·39-s + 1.18e4·41-s − 1.68e4·43-s − 1.30e4·47-s − 1.55e4·49-s + 2.13e3·51-s − 1.78e4·53-s + 9.68e3·57-s + 4.73e4·59-s − 2.27e4·61-s + 4.84e3·63-s − 3.94e4·67-s − 4.87e4·69-s + ⋯
L(s)  = 1  − 1.25·3-s + 0.270·7-s + 0.568·9-s + 1.06·11-s − 1.81·13-s − 0.0917·17-s − 0.315·19-s − 0.338·21-s + 0.984·23-s + 0.540·27-s − 0.00936·29-s + 1.49·31-s − 1.32·33-s + 1.65·37-s + 2.26·39-s + 1.10·41-s − 1.38·43-s − 0.860·47-s − 0.926·49-s + 0.114·51-s − 0.871·53-s + 0.394·57-s + 1.77·59-s − 0.783·61-s + 0.153·63-s − 1.07·67-s − 1.23·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ -1)\)

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 \( 1 + 19.5T + 243T^{2} \)
7 \( 1 - 35.0T + 1.68e4T^{2} \)
11 \( 1 - 426.T + 1.61e5T^{2} \)
13 \( 1 + 1.10e3T + 3.71e5T^{2} \)
17 \( 1 + 109.T + 1.41e6T^{2} \)
19 \( 1 + 495.T + 2.47e6T^{2} \)
23 \( 1 - 2.49e3T + 6.43e6T^{2} \)
29 \( 1 + 42.4T + 2.05e7T^{2} \)
31 \( 1 - 7.99e3T + 2.86e7T^{2} \)
37 \( 1 - 1.37e4T + 6.93e7T^{2} \)
41 \( 1 - 1.18e4T + 1.15e8T^{2} \)
43 \( 1 + 1.68e4T + 1.47e8T^{2} \)
47 \( 1 + 1.30e4T + 2.29e8T^{2} \)
53 \( 1 + 1.78e4T + 4.18e8T^{2} \)
59 \( 1 - 4.73e4T + 7.14e8T^{2} \)
61 \( 1 + 2.27e4T + 8.44e8T^{2} \)
67 \( 1 + 3.94e4T + 1.35e9T^{2} \)
71 \( 1 - 1.61e3T + 1.80e9T^{2} \)
73 \( 1 + 5.32e4T + 2.07e9T^{2} \)
79 \( 1 - 6.51e3T + 3.07e9T^{2} \)
83 \( 1 - 4.60e4T + 3.93e9T^{2} \)
89 \( 1 - 1.13e5T + 5.58e9T^{2} \)
97 \( 1 + 1.07e5T + 8.58e9T^{2} \)
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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

−10.07399825311794768999948364969, −9.315582684664104464562581782070, −8.037897245407251071222060169976, −6.91074543509366026083124771029, −6.23925707993129128822481199799, −5.05353689192378522833821747784, −4.44503600390338862130057311441, −2.72596894324336354677795325257, −1.18042055845482309029156719362, 0, 1.18042055845482309029156719362, 2.72596894324336354677795325257, 4.44503600390338862130057311441, 5.05353689192378522833821747784, 6.23925707993129128822481199799, 6.91074543509366026083124771029, 8.037897245407251071222060169976, 9.315582684664104464562581782070, 10.07399825311794768999948364969

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