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Properties

Label 2-142800-1.1-c1-0-88
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s − 4·13-s − 17-s − 6·19-s − 21-s − 7·23-s − 27-s − 2·29-s − 5·31-s + 33-s + 4·37-s + 4·39-s + 4·41-s + 3·47-s + 49-s + 51-s + 10·53-s + 6·57-s − 5·59-s + 13·61-s + 63-s + 3·67-s + 7·69-s − 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.242·17-s − 1.37·19-s − 0.218·21-s − 1.45·23-s − 0.192·27-s − 0.371·29-s − 0.898·31-s + 0.174·33-s + 0.657·37-s + 0.640·39-s + 0.624·41-s + 0.437·47-s + 1/7·49-s + 0.140·51-s + 1.37·53-s + 0.794·57-s − 0.650·59-s + 1.66·61-s + 0.125·63-s + 0.366·67-s + 0.842·69-s − 0.593·71-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: \(1\)
Selberg data: \((2,\ 142800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 + T \)
good11 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.52847559168416, −13.07229824377238, −12.61525646893073, −12.19103559447936, −11.75994606487494, −11.24087693455789, −10.71389417495744, −10.39061637154581, −9.837605388266919, −9.380111051830488, −8.758128557194084, −8.260134860311193, −7.623244385393262, −7.401441934687562, −6.636212712129899, −6.245954631657235, −5.601232721679995, −5.224432158564944, −4.585850520777699, −4.081322442510936, −3.693944206314431, −2.524411246162323, −2.294196082939457, −1.651566854618190, −0.6432133920667516, 0, 0.6432133920667516, 1.651566854618190, 2.294196082939457, 2.524411246162323, 3.693944206314431, 4.081322442510936, 4.585850520777699, 5.224432158564944, 5.601232721679995, 6.245954631657235, 6.636212712129899, 7.401441934687562, 7.623244385393262, 8.260134860311193, 8.758128557194084, 9.380111051830488, 9.837605388266919, 10.39061637154581, 10.71389417495744, 11.24087693455789, 11.75994606487494, 12.19103559447936, 12.61525646893073, 13.07229824377238, 13.52847559168416

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