Properties

Label 2-20e2-1.1-c5-0-1
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s − 88·7-s − 99·9-s − 540·11-s + 418·13-s − 594·17-s − 836·19-s + 1.05e3·21-s − 4.10e3·23-s + 4.10e3·27-s − 594·29-s − 4.25e3·31-s + 6.48e3·33-s + 298·37-s − 5.01e3·39-s + 1.72e4·41-s − 1.21e4·43-s − 1.29e3·47-s − 9.06e3·49-s + 7.12e3·51-s − 1.94e4·53-s + 1.00e4·57-s + 7.66e3·59-s − 3.47e4·61-s + 8.71e3·63-s + 2.18e4·67-s + 4.92e4·69-s + ⋯
L(s)  = 1  − 0.769·3-s − 0.678·7-s − 0.407·9-s − 1.34·11-s + 0.685·13-s − 0.498·17-s − 0.531·19-s + 0.522·21-s − 1.61·23-s + 1.08·27-s − 0.131·29-s − 0.795·31-s + 1.03·33-s + 0.0357·37-s − 0.528·39-s + 1.60·41-s − 0.997·43-s − 0.0855·47-s − 0.539·49-s + 0.383·51-s − 0.953·53-s + 0.408·57-s + 0.286·59-s − 1.19·61-s + 0.276·63-s + 0.593·67-s + 1.24·69-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

\(L(3)\) \(\approx\) \(0.4951380905\)
\(L(\frac12)\) \(\approx\) \(0.4951380905\)
\(L(\frac{7}{2})\) 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 \( 1 + 4 p T + p^{5} T^{2} \)
7 \( 1 + 88 T + p^{5} T^{2} \)
11 \( 1 + 540 T + p^{5} T^{2} \)
13 \( 1 - 418 T + p^{5} T^{2} \)
17 \( 1 + 594 T + p^{5} T^{2} \)
19 \( 1 + 44 p T + p^{5} T^{2} \)
23 \( 1 + 4104 T + p^{5} T^{2} \)
29 \( 1 + 594 T + p^{5} T^{2} \)
31 \( 1 + 4256 T + p^{5} T^{2} \)
37 \( 1 - 298 T + p^{5} T^{2} \)
41 \( 1 - 17226 T + p^{5} T^{2} \)
43 \( 1 + 12100 T + p^{5} T^{2} \)
47 \( 1 + 1296 T + p^{5} T^{2} \)
53 \( 1 + 19494 T + p^{5} T^{2} \)
59 \( 1 - 7668 T + p^{5} T^{2} \)
61 \( 1 + 34738 T + p^{5} T^{2} \)
67 \( 1 - 21812 T + p^{5} T^{2} \)
71 \( 1 - 46872 T + p^{5} T^{2} \)
73 \( 1 + 67562 T + p^{5} T^{2} \)
79 \( 1 - 76912 T + p^{5} T^{2} \)
83 \( 1 - 67716 T + p^{5} T^{2} \)
89 \( 1 - 29754 T + p^{5} T^{2} \)
97 \( 1 - 122398 T + p^{5} T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64904729172115075975351157055, −9.703144930712853659992587894501, −8.574672049114380686164620451120, −7.69272570469616365474407029740, −6.36966168520831904151986313229, −5.84012022924798646875943307799, −4.74613222887416704278550168572, −3.42039788671378080144164335057, −2.17475669847966030665475965789, −0.36361852741090987730834854544, 0.36361852741090987730834854544, 2.17475669847966030665475965789, 3.42039788671378080144164335057, 4.74613222887416704278550168572, 5.84012022924798646875943307799, 6.36966168520831904151986313229, 7.69272570469616365474407029740, 8.574672049114380686164620451120, 9.703144930712853659992587894501, 10.64904729172115075975351157055

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