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Properties

Label 2-142800-1.1-c1-0-177
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 − 2·11-s + 5·13-s − 17-s − 2·19-s − 21-s − 23-s − 27-s + 8·29-s − 31-s + 2·33-s + 3·37-s − 5·39-s − 7·41-s − 47-s + 49-s + 51-s + 8·53-s + 2·57-s − 7·61-s + 63-s + 16·67-s + 69-s + 10·71-s + 8·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 1.38·13-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.48·29-s − 0.179·31-s + 0.348·33-s + 0.493·37-s − 0.800·39-s − 1.09·41-s − 0.145·47-s + 1/7·49-s + 0.140·51-s + 1.09·53-s + 0.264·57-s − 0.896·61-s + 0.125·63-s + 1.95·67-s + 0.120·69-s + 1.18·71-s + 0.936·73-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.67053947313673, −13.12866383892184, −12.66658508875675, −12.17946520860363, −11.68598147494558, −11.10162939704288, −10.91336864016891, −10.32479963176063, −9.955467440847751, −9.288209495580561, −8.629926224784929, −8.303296821965458, −7.912274208481541, −7.172848475999406, −6.563020890996857, −6.326597043695587, −5.667101013661405, −5.111726553147115, −4.741290005156871, −3.938138312416631, −3.671674196145129, −2.742579316822733, −2.229728565671606, −1.404462665969095, −0.8887999832685159, 0, 0.8887999832685159, 1.404462665969095, 2.229728565671606, 2.742579316822733, 3.671674196145129, 3.938138312416631, 4.741290005156871, 5.111726553147115, 5.667101013661405, 6.326597043695587, 6.563020890996857, 7.172848475999406, 7.912274208481541, 8.303296821965458, 8.629926224784929, 9.288209495580561, 9.955467440847751, 10.32479963176063, 10.91336864016891, 11.10162939704288, 11.68598147494558, 12.17946520860363, 12.66658508875675, 13.12866383892184, 13.67053947313673

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