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Properties

Label 2-142800-1.1-c1-0-128
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 + 6·11-s − 17-s + 6·19-s + 21-s + 4·23-s + 27-s − 4·29-s + 8·31-s + 6·33-s − 4·37-s − 2·41-s + 12·43-s − 8·47-s + 49-s − 51-s + 6·53-s + 6·57-s + 14·59-s + 2·61-s + 63-s − 8·67-s + 4·69-s + 6·73-s + 6·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.242·17-s + 1.37·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s − 0.742·29-s + 1.43·31-s + 1.04·33-s − 0.657·37-s − 0.312·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.140·51-s + 0.824·53-s + 0.794·57-s + 1.82·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s + 0.481·69-s + 0.702·73-s + 0.683·77-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

\(L(1)\) \(\approx\) \(5.619520735\)
\(L(\frac12)\) \(\approx\) \(5.619520735\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.49837543443678, −12.96664834052166, −12.43793202864083, −11.77387613553635, −11.60189552298589, −11.18919431197762, −10.38258440371194, −9.965979303505523, −9.424937016189014, −8.984783384566521, −8.704362936321996, −8.072657783291627, −7.434142122168014, −7.094947423093606, −6.573362661222806, −6.010519317874813, −5.371742818112034, −4.777416033236760, −4.250001517575245, −3.668862852259869, −3.264777601512643, −2.530991517377462, −1.869213711327504, −1.171734672240454, −0.7792454771617319, 0.7792454771617319, 1.171734672240454, 1.869213711327504, 2.530991517377462, 3.264777601512643, 3.668862852259869, 4.250001517575245, 4.777416033236760, 5.371742818112034, 6.010519317874813, 6.573362661222806, 7.094947423093606, 7.434142122168014, 8.072657783291627, 8.704362936321996, 8.984783384566521, 9.424937016189014, 9.965979303505523, 10.38258440371194, 11.18919431197762, 11.60189552298589, 11.77387613553635, 12.43793202864083, 12.96664834052166, 13.49837543443678

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