Properties

Label 2-64400-1.1-c1-0-57
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 + 3·11-s + 3·13-s + 3·17-s + 2·19-s − 21-s − 23-s + 5·27-s + 3·29-s + 4·31-s − 3·33-s + 8·37-s − 3·39-s − 10·41-s − 4·43-s − 13·47-s + 49-s − 3·51-s − 2·57-s + 8·59-s − 10·61-s − 2·63-s + 6·67-s + 69-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s + 0.727·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s − 0.609·43-s − 1.89·47-s + 1/7·49-s − 0.420·51-s − 0.264·57-s + 1.04·59-s − 1.28·61-s − 0.251·63-s + 0.733·67-s + 0.120·69-s + 0.702·73-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: \(1\)
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 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.52628452165139, −14.00195049950482, −13.52785965983604, −13.02674130586958, −12.23260721496235, −11.87881974973279, −11.52324673127664, −11.08782873863068, −10.53303358539207, −9.782833832300758, −9.605308933323171, −8.608106917275398, −8.415447252086902, −7.910485692428081, −7.090571590438462, −6.472416560511175, −6.193361920109103, −5.529463124573457, −4.993398524990312, −4.454404870918813, −3.633289753004798, −3.214834984590904, −2.434934019731653, −1.410854248109441, −1.065398670791345, 0, 1.065398670791345, 1.410854248109441, 2.434934019731653, 3.214834984590904, 3.633289753004798, 4.454404870918813, 4.993398524990312, 5.529463124573457, 6.193361920109103, 6.472416560511175, 7.090571590438462, 7.910485692428081, 8.415447252086902, 8.608106917275398, 9.605308933323171, 9.782833832300758, 10.53303358539207, 11.08782873863068, 11.52324673127664, 11.87881974973279, 12.23260721496235, 13.02674130586958, 13.52785965983604, 14.00195049950482, 14.52628452165139

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