Properties

Label 2-64400-1.1-c1-0-0
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 − 6·13-s − 6·17-s − 2·21-s + 23-s − 4·27-s + 2·29-s − 8·31-s − 4·33-s + 2·37-s − 12·39-s − 2·41-s + 4·43-s + 6·47-s + 49-s − 12·51-s − 14·53-s + 12·59-s − 63-s − 12·67-s + 2·69-s + 12·71-s − 6·73-s + 2·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 1.45·17-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 0.328·37-s − 1.92·39-s − 0.312·41-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 1.68·51-s − 1.92·53-s + 1.56·59-s − 0.125·63-s − 1.46·67-s + 0.240·69-s + 1.42·71-s − 0.702·73-s + 0.227·77-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

\(L(1)\) \(\approx\) \(0.7694169050\)
\(L(\frac12)\) \(\approx\) \(0.7694169050\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−14.23132197731012, −13.85924577682844, −13.08995740725573, −12.95512446635738, −12.41142807351176, −11.72500490963125, −11.15365110857884, −10.65006668147077, −9.995232006705254, −9.542796433387516, −9.109386948962753, −8.658777916567256, −8.050246577332205, −7.524974095724326, −7.081473117243959, −6.594036023228363, −5.668645095907518, −5.271097743743150, −4.416723112364775, −4.110448915701149, −3.154472474286590, −2.719049106613662, −2.298600372126727, −1.621969659927204, −0.2494477513874447, 0.2494477513874447, 1.621969659927204, 2.298600372126727, 2.719049106613662, 3.154472474286590, 4.110448915701149, 4.416723112364775, 5.271097743743150, 5.668645095907518, 6.594036023228363, 7.081473117243959, 7.524974095724326, 8.050246577332205, 8.658777916567256, 9.109386948962753, 9.542796433387516, 9.995232006705254, 10.65006668147077, 11.15365110857884, 11.72500490963125, 12.41142807351176, 12.95512446635738, 13.08995740725573, 13.85924577682844, 14.23132197731012

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