Properties

Degree $2$
Conductor $400$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.5·3-s + 131.·7-s + 408.·9-s − 290.·11-s − 68.3·13-s + 310.·17-s + 2.13e3·19-s + 3.34e3·21-s + 873.·23-s + 4.22e3·27-s − 2.58e3·29-s + 9.08e3·31-s − 7.40e3·33-s − 3.99e3·37-s − 1.74e3·39-s + 1.69e4·41-s − 1.80e4·43-s + 2.48e4·47-s + 366.·49-s + 7.92e3·51-s + 7.65e3·53-s + 5.44e4·57-s + 9.23e3·59-s + 3.32e3·61-s + 5.35e4·63-s − 3.23e4·67-s + 2.22e4·69-s + ⋯
L(s)  = 1  + 1.63·3-s + 1.01·7-s + 1.68·9-s − 0.722·11-s − 0.112·13-s + 0.260·17-s + 1.35·19-s + 1.65·21-s + 0.344·23-s + 1.11·27-s − 0.569·29-s + 1.69·31-s − 1.18·33-s − 0.479·37-s − 0.183·39-s + 1.57·41-s − 1.48·43-s + 1.64·47-s + 0.0218·49-s + 0.426·51-s + 0.374·53-s + 2.21·57-s + 0.345·59-s + 0.114·61-s + 1.69·63-s − 0.880·67-s + 0.563·69-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: $1$
Motivic weight: \(5\)
Character: $\chi_{400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.675817421\)
\(L(\frac12)\) \(\approx\) \(4.675817421\)
\(L(\frac{7}{2})\) 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 \( 1 - 25.5T + 243T^{2} \)
7 \( 1 - 131.T + 1.68e4T^{2} \)
11 \( 1 + 290.T + 1.61e5T^{2} \)
13 \( 1 + 68.3T + 3.71e5T^{2} \)
17 \( 1 - 310.T + 1.41e6T^{2} \)
19 \( 1 - 2.13e3T + 2.47e6T^{2} \)
23 \( 1 - 873.T + 6.43e6T^{2} \)
29 \( 1 + 2.58e3T + 2.05e7T^{2} \)
31 \( 1 - 9.08e3T + 2.86e7T^{2} \)
37 \( 1 + 3.99e3T + 6.93e7T^{2} \)
41 \( 1 - 1.69e4T + 1.15e8T^{2} \)
43 \( 1 + 1.80e4T + 1.47e8T^{2} \)
47 \( 1 - 2.48e4T + 2.29e8T^{2} \)
53 \( 1 - 7.65e3T + 4.18e8T^{2} \)
59 \( 1 - 9.23e3T + 7.14e8T^{2} \)
61 \( 1 - 3.32e3T + 8.44e8T^{2} \)
67 \( 1 + 3.23e4T + 1.35e9T^{2} \)
71 \( 1 - 3.58e4T + 1.80e9T^{2} \)
73 \( 1 - 2.65e4T + 2.07e9T^{2} \)
79 \( 1 + 7.17e4T + 3.07e9T^{2} \)
83 \( 1 - 3.96e4T + 3.93e9T^{2} \)
89 \( 1 + 1.17e5T + 5.58e9T^{2} \)
97 \( 1 + 2.18e4T + 8.58e9T^{2} \)
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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

−10.21761518961975166188871211990, −9.427370150805717267392975804051, −8.475789529566974499214828139504, −7.86731576539804107883479821291, −7.17305070309182479320245476207, −5.46096694243240174390212588375, −4.41076396743122626779521326921, −3.20797351358336802516694669401, −2.33922417839937057309173227973, −1.15923660263639035114368770645, 1.15923660263639035114368770645, 2.33922417839937057309173227973, 3.20797351358336802516694669401, 4.41076396743122626779521326921, 5.46096694243240174390212588375, 7.17305070309182479320245476207, 7.86731576539804107883479821291, 8.475789529566974499214828139504, 9.427370150805717267392975804051, 10.21761518961975166188871211990

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