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Properties

Label 2-142800-1.1-c1-0-81
Degree $2$
Conductor $142800$
Sign $1$
Analytic cond. $1140.26$
Root an. cond. $33.7677$
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 + 9-s + 2·13-s + 17-s + 8·19-s − 21-s + 4·23-s − 27-s + 2·29-s + 2·37-s − 2·39-s − 10·41-s + 4·43-s − 8·47-s + 49-s − 51-s + 10·53-s − 8·57-s − 4·59-s + 10·61-s + 63-s − 4·67-s − 4·69-s + 12·71-s + 10·73-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.242·17-s + 1.83·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s + 0.328·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 1.37·53-s − 1.05·57-s − 0.520·59-s + 1.28·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s + 1.42·71-s + 1.17·73-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(142800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(1140.26\)
Root analytic conductor: \(33.7677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 142800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.946807533\)
\(L(\frac12)\) \(\approx\) \(2.946807533\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.42853011604951, −12.95501158187243, −12.27791838192024, −11.94169579144634, −11.38850663684623, −11.21702270601015, −10.46952651247783, −10.14170392557094, −9.537440465773395, −9.119375470507596, −8.516248016156247, −7.921715576349407, −7.597450965359646, −6.756804384506004, −6.706294468910716, −5.832103286296285, −5.272335892330101, −5.113962890281672, −4.396665007088287, −3.655802130638829, −3.278445628126769, −2.549793711912573, −1.734235194080112, −1.091830686294439, −0.6216530268942767, 0.6216530268942767, 1.091830686294439, 1.734235194080112, 2.549793711912573, 3.278445628126769, 3.655802130638829, 4.396665007088287, 5.113962890281672, 5.272335892330101, 5.832103286296285, 6.706294468910716, 6.756804384506004, 7.597450965359646, 7.921715576349407, 8.516248016156247, 9.119375470507596, 9.537440465773395, 10.14170392557094, 10.46952651247783, 11.21702270601015, 11.38850663684623, 11.94169579144634, 12.27791838192024, 12.95501158187243, 13.42853011604951

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