Properties

Label 2-20e2-1.1-c5-0-13
Degree $2$
Conductor $400$
Sign $1$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 192·7-s − 227·9-s + 148·11-s − 286·13-s + 1.67e3·17-s − 1.06e3·19-s − 768·21-s + 2.97e3·23-s + 1.88e3·27-s − 3.41e3·29-s + 2.44e3·31-s − 592·33-s − 182·37-s + 1.14e3·39-s − 9.39e3·41-s − 1.24e3·43-s − 1.20e4·47-s + 2.00e4·49-s − 6.71e3·51-s − 2.38e4·53-s + 4.24e3·57-s + 2.00e4·59-s + 3.23e4·61-s − 4.35e4·63-s + 6.09e4·67-s − 1.19e4·69-s + ⋯
L(s)  = 1  − 0.256·3-s + 1.48·7-s − 0.934·9-s + 0.368·11-s − 0.469·13-s + 1.40·17-s − 0.673·19-s − 0.380·21-s + 1.17·23-s + 0.496·27-s − 0.752·29-s + 0.457·31-s − 0.0946·33-s − 0.0218·37-s + 0.120·39-s − 0.873·41-s − 0.102·43-s − 0.798·47-s + 1.19·49-s − 0.361·51-s − 1.16·53-s + 0.172·57-s + 0.748·59-s + 1.11·61-s − 1.38·63-s + 1.65·67-s − 0.301·69-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: yes
Arithmetic: yes
Character: $\chi_{400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.193792816\)
\(L(\frac12)\) \(\approx\) \(2.193792816\)
\(L(\frac{7}{2})\) 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 \( 1 + 4 T + p^{5} T^{2} \)
7 \( 1 - 192 T + p^{5} T^{2} \)
11 \( 1 - 148 T + p^{5} T^{2} \)
13 \( 1 + 22 p T + p^{5} T^{2} \)
17 \( 1 - 1678 T + p^{5} T^{2} \)
19 \( 1 + 1060 T + p^{5} T^{2} \)
23 \( 1 - 2976 T + p^{5} T^{2} \)
29 \( 1 + 3410 T + p^{5} T^{2} \)
31 \( 1 - 2448 T + p^{5} T^{2} \)
37 \( 1 + 182 T + p^{5} T^{2} \)
41 \( 1 + 9398 T + p^{5} T^{2} \)
43 \( 1 + 1244 T + p^{5} T^{2} \)
47 \( 1 + 12088 T + p^{5} T^{2} \)
53 \( 1 + 23846 T + p^{5} T^{2} \)
59 \( 1 - 20020 T + p^{5} T^{2} \)
61 \( 1 - 32302 T + p^{5} T^{2} \)
67 \( 1 - 60972 T + p^{5} T^{2} \)
71 \( 1 - 32648 T + p^{5} T^{2} \)
73 \( 1 - 38774 T + p^{5} T^{2} \)
79 \( 1 - 33360 T + p^{5} T^{2} \)
83 \( 1 - 16716 T + p^{5} T^{2} \)
89 \( 1 - 101370 T + p^{5} T^{2} \)
97 \( 1 - 119038 T + p^{5} T^{2} \)
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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

−10.67027666058708163738462207637, −9.532705544054512927758738073920, −8.450080944333447225857978324268, −7.86229899937464361653769540228, −6.67844691001446821989036432789, −5.43284793851267509836559124010, −4.85634525509475463013255653711, −3.42446589288220369190773473550, −2.04438800696120404042305604576, −0.815420413783544868449753272062, 0.815420413783544868449753272062, 2.04438800696120404042305604576, 3.42446589288220369190773473550, 4.85634525509475463013255653711, 5.43284793851267509836559124010, 6.67844691001446821989036432789, 7.86229899937464361653769540228, 8.450080944333447225857978324268, 9.532705544054512927758738073920, 10.67027666058708163738462207637

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