Properties

Label 2-64400-1.1-c1-0-22
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 + 6·11-s + 4·13-s − 6·17-s − 8·19-s − 2·21-s + 23-s + 4·27-s + 6·29-s + 4·31-s − 12·33-s + 4·37-s − 8·39-s + 6·41-s − 10·43-s + 12·47-s + 49-s + 12·51-s + 16·57-s + 6·59-s + 14·61-s + 63-s + 14·67-s − 2·69-s − 14·73-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s − 1.45·17-s − 1.83·19-s − 0.436·21-s + 0.208·23-s + 0.769·27-s + 1.11·29-s + 0.718·31-s − 2.08·33-s + 0.657·37-s − 1.28·39-s + 0.937·41-s − 1.52·43-s + 1.75·47-s + 1/7·49-s + 1.68·51-s + 2.11·57-s + 0.781·59-s + 1.79·61-s + 0.125·63-s + 1.71·67-s − 0.240·69-s − 1.63·73-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956276200\)
\(L(\frac12)\) \(\approx\) \(1.956276200\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−14.23250796192688, −13.71641399271573, −13.18469883217535, −12.58710877950345, −12.16860322602902, −11.48134851793285, −11.29295864679163, −10.93663097040838, −10.31010411039955, −9.763108960738974, −8.893527277767019, −8.537176433173033, −8.424846445364771, −7.189874064050129, −6.675059298530701, −6.345506055484596, −6.060752375622995, −5.270754391427104, −4.443990051474045, −4.287292683366526, −3.682463748880848, −2.608777373891628, −1.960327546365395, −1.102590764620942, −0.5992728275760118, 0.5992728275760118, 1.102590764620942, 1.960327546365395, 2.608777373891628, 3.682463748880848, 4.287292683366526, 4.443990051474045, 5.270754391427104, 6.060752375622995, 6.345506055484596, 6.675059298530701, 7.189874064050129, 8.424846445364771, 8.537176433173033, 8.893527277767019, 9.763108960738974, 10.31010411039955, 10.93663097040838, 11.29295864679163, 11.48134851793285, 12.16860322602902, 12.58710877950345, 13.18469883217535, 13.71641399271573, 14.23250796192688

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