Properties

Label 2-64400-1.1-c1-0-37
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  − 3-s − 7-s − 2·9-s + 5·11-s + 2·13-s − 5·17-s − 5·19-s + 21-s − 23-s + 5·27-s − 2·29-s + 4·31-s − 5·33-s + 4·37-s − 2·39-s − 5·41-s + 8·43-s + 6·47-s + 49-s + 5·51-s − 4·53-s + 5·57-s − 4·59-s + 10·61-s + 2·63-s − 67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s + 0.554·13-s − 1.21·17-s − 1.14·19-s + 0.218·21-s − 0.208·23-s + 0.962·27-s − 0.371·29-s + 0.718·31-s − 0.870·33-s + 0.657·37-s − 0.320·39-s − 0.780·41-s + 1.21·43-s + 0.875·47-s + 1/7·49-s + 0.700·51-s − 0.549·53-s + 0.662·57-s − 0.520·59-s + 1.28·61-s + 0.251·63-s − 0.122·67-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: \(1\)
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 \( 1 + T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.43773854913297, −14.02884601208600, −13.46037864029918, −12.94206321573453, −12.44104848380303, −11.81317009165413, −11.54164226735065, −10.95442563411833, −10.62752380254706, −9.931693526898490, −9.217337746521572, −8.910242633319014, −8.472358037901858, −7.823788266819982, −6.834223414889623, −6.701579317307176, −6.041139929826834, −5.810625377648077, −4.891592402693916, −4.155470066939465, −4.013999438669239, −3.052302948618251, −2.421984224368438, −1.642764891864162, −0.8197933952858737, 0, 0.8197933952858737, 1.642764891864162, 2.421984224368438, 3.052302948618251, 4.013999438669239, 4.155470066939465, 4.891592402693916, 5.810625377648077, 6.041139929826834, 6.701579317307176, 6.834223414889623, 7.823788266819982, 8.472358037901858, 8.910242633319014, 9.217337746521572, 9.931693526898490, 10.62752380254706, 10.95442563411833, 11.54164226735065, 11.81317009165413, 12.44104848380303, 12.94206321573453, 13.46037864029918, 14.02884601208600, 14.43773854913297

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