Properties

Label 2-64400-1.1-c1-0-13
Degree $2$
Conductor $64400$
Sign $1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 2·11-s − 5·13-s + 5·17-s + 4·19-s − 2·21-s − 23-s + 4·27-s + 2·29-s + 4·33-s + 9·37-s + 10·39-s + 4·43-s + 47-s + 49-s − 10·51-s − 5·53-s − 8·57-s + 12·59-s + 4·61-s + 63-s − 4·67-s + 2·69-s − 2·71-s + 2·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 1.21·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 0.371·29-s + 0.696·33-s + 1.47·37-s + 1.60·39-s + 0.609·43-s + 0.145·47-s + 1/7·49-s − 1.40·51-s − 0.686·53-s − 1.05·57-s + 1.56·59-s + 0.512·61-s + 0.125·63-s − 0.488·67-s + 0.240·69-s − 0.237·71-s + 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445282464\)
\(L(\frac12)\) \(\approx\) \(1.445282464\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 - 13 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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

−14.45902747700621, −13.72448948638372, −13.11670971236557, −12.54071821693744, −12.09310493460612, −11.79730499237113, −11.29604497530846, −10.68120576222467, −10.28352186023504, −9.665742583848707, −9.421303080285737, −8.391913956218177, −7.972015083052386, −7.409212184356695, −7.023362581919559, −6.224675543483602, −5.639116233358934, −5.356888684182870, −4.773959793262904, −4.312431147733085, −3.317078113358673, −2.751299498741644, −2.068310877787747, −1.027586841866586, −0.5253158872978368, 0.5253158872978368, 1.027586841866586, 2.068310877787747, 2.751299498741644, 3.317078113358673, 4.312431147733085, 4.773959793262904, 5.356888684182870, 5.639116233358934, 6.224675543483602, 7.023362581919559, 7.409212184356695, 7.972015083052386, 8.391913956218177, 9.421303080285737, 9.665742583848707, 10.28352186023504, 10.68120576222467, 11.29604497530846, 11.79730499237113, 12.09310493460612, 12.54071821693744, 13.11670971236557, 13.72448948638372, 14.45902747700621

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