Properties

Label 2-64400-1.1-c1-0-18
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s − 2·9-s + 11-s + 13-s + 17-s − 2·19-s − 21-s − 23-s + 5·27-s + 7·29-s − 4·31-s − 33-s + 8·37-s − 39-s − 6·41-s + 8·43-s + 7·47-s + 49-s − 51-s + 4·53-s + 2·57-s + 4·59-s − 10·61-s − 2·63-s + 14·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s + 1.29·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.160·39-s − 0.937·41-s + 1.21·43-s + 1.02·47-s + 1/7·49-s − 0.140·51-s + 0.549·53-s + 0.264·57-s + 0.520·59-s − 1.28·61-s − 0.251·63-s + 1.71·67-s + 0.120·69-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

\(L(1)\) \(\approx\) \(1.984005431\)
\(L(\frac12)\) \(\approx\) \(1.984005431\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.23711736124725, −13.77254609466009, −13.29281678298830, −12.54665612132162, −12.12208034646403, −11.80947691487956, −11.13541287007282, −10.72686329264173, −10.43767172746399, −9.514334110865576, −9.210161118375902, −8.469339380081694, −8.124665723352248, −7.556074743879272, −6.691315513413977, −6.462750522186132, −5.696867744706072, −5.391700986341262, −4.650119490303055, −4.124213523339483, −3.455556815802171, −2.668048971507700, −2.127525913765752, −1.134436370341860, −0.5593200447105998, 0.5593200447105998, 1.134436370341860, 2.127525913765752, 2.668048971507700, 3.455556815802171, 4.124213523339483, 4.650119490303055, 5.391700986341262, 5.696867744706072, 6.462750522186132, 6.691315513413977, 7.556074743879272, 8.124665723352248, 8.469339380081694, 9.210161118375902, 9.514334110865576, 10.43767172746399, 10.72686329264173, 11.13541287007282, 11.80947691487956, 12.12208034646403, 12.54665612132162, 13.29281678298830, 13.77254609466009, 14.23711736124725

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