Properties

Label 2-64400-1.1-c1-0-44
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 2·13-s + 6·17-s + 4·19-s − 2·21-s − 23-s + 4·27-s + 6·29-s − 2·31-s + 10·37-s + 4·39-s − 6·41-s − 10·43-s + 49-s − 12·51-s − 6·53-s − 8·57-s + 12·59-s − 10·61-s + 63-s + 2·67-s + 2·69-s − 12·71-s + 16·73-s + 10·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 1.11·29-s − 0.359·31-s + 1.64·37-s + 0.640·39-s − 0.937·41-s − 1.52·43-s + 1/7·49-s − 1.68·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.125·63-s + 0.244·67-s + 0.240·69-s − 1.42·71-s + 1.87·73-s + 1.12·79-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: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.38775632971249, −14.09281108965422, −13.49623325892344, −12.76588524091293, −12.34994123324681, −11.88861666616304, −11.49947368347028, −11.15472713190597, −10.33151086759935, −10.03151320563821, −9.639726470872890, −8.800289700352483, −8.231635928769068, −7.680271694567016, −7.251313792363088, −6.458672832629183, −6.128199724146251, −5.397983957261981, −5.054867131432142, −4.639677670561611, −3.716068081186972, −3.116830325813911, −2.436116004842388, −1.409104599745157, −0.9194585274592567, 0, 0.9194585274592567, 1.409104599745157, 2.436116004842388, 3.116830325813911, 3.716068081186972, 4.639677670561611, 5.054867131432142, 5.397983957261981, 6.128199724146251, 6.458672832629183, 7.251313792363088, 7.680271694567016, 8.231635928769068, 8.800289700352483, 9.639726470872890, 10.03151320563821, 10.33151086759935, 11.15472713190597, 11.49947368347028, 11.88861666616304, 12.34994123324681, 12.76588524091293, 13.49623325892344, 14.09281108965422, 14.38775632971249

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