Properties

Label 2-20e2-1.1-c5-0-37
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  + 11.5·3-s − 27.0·7-s − 110.·9-s − 226.·11-s + 511.·13-s + 387.·17-s + 1.33e3·19-s − 311.·21-s − 545.·23-s − 4.07e3·27-s − 4.63e3·29-s − 2.99e3·31-s − 2.60e3·33-s − 1.26e3·37-s + 5.89e3·39-s − 1.71e4·41-s + 1.65e4·43-s − 1.30e4·47-s − 1.60e4·49-s + 4.46e3·51-s − 2.89e4·53-s + 1.53e4·57-s + 3.44e4·59-s − 2.41e4·61-s + 2.98e3·63-s − 2.93e4·67-s − 6.28e3·69-s + ⋯
L(s)  = 1  + 0.739·3-s − 0.208·7-s − 0.453·9-s − 0.563·11-s + 0.839·13-s + 0.325·17-s + 0.848·19-s − 0.154·21-s − 0.214·23-s − 1.07·27-s − 1.02·29-s − 0.559·31-s − 0.416·33-s − 0.151·37-s + 0.620·39-s − 1.59·41-s + 1.36·43-s − 0.860·47-s − 0.956·49-s + 0.240·51-s − 1.41·53-s + 0.627·57-s + 1.28·59-s − 0.830·61-s + 0.0946·63-s − 0.799·67-s − 0.158·69-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: $\chi_{400} (1, \cdot )$
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 \( 1 - 11.5T + 243T^{2} \)
7 \( 1 + 27.0T + 1.68e4T^{2} \)
11 \( 1 + 226.T + 1.61e5T^{2} \)
13 \( 1 - 511.T + 3.71e5T^{2} \)
17 \( 1 - 387.T + 1.41e6T^{2} \)
19 \( 1 - 1.33e3T + 2.47e6T^{2} \)
23 \( 1 + 545.T + 6.43e6T^{2} \)
29 \( 1 + 4.63e3T + 2.05e7T^{2} \)
31 \( 1 + 2.99e3T + 2.86e7T^{2} \)
37 \( 1 + 1.26e3T + 6.93e7T^{2} \)
41 \( 1 + 1.71e4T + 1.15e8T^{2} \)
43 \( 1 - 1.65e4T + 1.47e8T^{2} \)
47 \( 1 + 1.30e4T + 2.29e8T^{2} \)
53 \( 1 + 2.89e4T + 4.18e8T^{2} \)
59 \( 1 - 3.44e4T + 7.14e8T^{2} \)
61 \( 1 + 2.41e4T + 8.44e8T^{2} \)
67 \( 1 + 2.93e4T + 1.35e9T^{2} \)
71 \( 1 + 9.06e3T + 1.80e9T^{2} \)
73 \( 1 + 5.55e4T + 2.07e9T^{2} \)
79 \( 1 - 1.01e5T + 3.07e9T^{2} \)
83 \( 1 + 7.32e4T + 3.93e9T^{2} \)
89 \( 1 + 4.24e4T + 5.58e9T^{2} \)
97 \( 1 - 1.05e4T + 8.58e9T^{2} \)
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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

−9.850050317417212823167231161776, −9.045691449675921951607111563744, −8.185213220029995711251466559035, −7.42147561562159808907983681689, −6.12161321556378951404977117395, −5.20732936628718263341057791395, −3.70270476818462225555822481114, −2.94310999864192487169989714010, −1.62694506984968860545250238413, 0, 1.62694506984968860545250238413, 2.94310999864192487169989714010, 3.70270476818462225555822481114, 5.20732936628718263341057791395, 6.12161321556378951404977117395, 7.42147561562159808907983681689, 8.185213220029995711251466559035, 9.045691449675921951607111563744, 9.850050317417212823167231161776

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