Properties

Label 2-64400-1.1-c1-0-24
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  − 3·3-s − 7-s + 6·9-s + 6·11-s − 13-s + 3·21-s + 23-s − 9·27-s − 3·29-s + 3·31-s − 18·33-s + 8·37-s + 3·39-s + 9·41-s + 4·43-s + 13·47-s + 49-s − 4·53-s − 4·59-s + 2·61-s − 6·63-s − 4·67-s − 3·69-s + 5·71-s − 3·73-s − 6·77-s − 12·79-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s + 1.80·11-s − 0.277·13-s + 0.654·21-s + 0.208·23-s − 1.73·27-s − 0.557·29-s + 0.538·31-s − 3.13·33-s + 1.31·37-s + 0.480·39-s + 1.40·41-s + 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.549·53-s − 0.520·59-s + 0.256·61-s − 0.755·63-s − 0.488·67-s − 0.361·69-s + 0.593·71-s − 0.351·73-s − 0.683·77-s − 1.35·79-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

\(L(1)\) \(\approx\) \(1.610131014\)
\(L(\frac12)\) \(\approx\) \(1.610131014\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.24282093135259, −13.72123376617504, −12.92025527535809, −12.65757283186855, −12.06339272204154, −11.75624293705140, −11.25615143758669, −10.84779089543667, −10.27569682345686, −9.692003649189914, −9.210436854805638, −8.858348449133641, −7.722357008612807, −7.390754281382156, −6.706610660597112, −6.284022305632861, −5.938980935406601, −5.394412743708508, −4.533378801535727, −4.272802348339132, −3.668819410857720, −2.724791560912354, −1.824095076391545, −0.9866521422317045, −0.6193452208635697, 0.6193452208635697, 0.9866521422317045, 1.824095076391545, 2.724791560912354, 3.668819410857720, 4.272802348339132, 4.533378801535727, 5.394412743708508, 5.938980935406601, 6.284022305632861, 6.706610660597112, 7.390754281382156, 7.722357008612807, 8.858348449133641, 9.210436854805638, 9.692003649189914, 10.27569682345686, 10.84779089543667, 11.25615143758669, 11.75624293705140, 12.06339272204154, 12.65757283186855, 12.92025527535809, 13.72123376617504, 14.24282093135259

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