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Properties

Label 2-193200-1.1-c1-0-111
Degree $2$
Conductor $193200$
Sign $-1$
Analytic cond. $1542.70$
Root an. cond. $39.2773$
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 + 4·19-s − 21-s − 23-s + 27-s − 6·29-s − 2·31-s + 33-s + 6·37-s − 4·39-s − 9·41-s − 2·43-s − 8·47-s + 49-s + 53-s + 4·57-s + 59-s − 61-s − 63-s + 8·67-s − 69-s − 6·71-s + 12·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.917·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.174·33-s + 0.986·37-s − 0.640·39-s − 1.40·41-s − 0.304·43-s − 1.16·47-s + 1/7·49-s + 0.137·53-s + 0.529·57-s + 0.130·59-s − 0.128·61-s − 0.125·63-s + 0.977·67-s − 0.120·69-s − 0.712·71-s + 1.40·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(193200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(1542.70\)
Root analytic conductor: \(39.2773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 193200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−13.31561883419425, −12.90150328179575, −12.47762284461594, −11.91569266145479, −11.49376993580503, −11.10412096337783, −10.29313547688341, −9.900159174019069, −9.609514739627184, −9.175232860164530, −8.551403102538024, −8.134961425324152, −7.406278574687697, −7.303132491137431, −6.675654057326518, −6.068858002809635, −5.536757615808755, −4.893163282068347, −4.562782175201971, −3.671112202992686, −3.433978774515922, −2.806538003592021, −2.129417676082443, −1.683638890805328, −0.7980030356845892, 0, 0.7980030356845892, 1.683638890805328, 2.129417676082443, 2.806538003592021, 3.433978774515922, 3.671112202992686, 4.562782175201971, 4.893163282068347, 5.536757615808755, 6.068858002809635, 6.675654057326518, 7.303132491137431, 7.406278574687697, 8.134961425324152, 8.551403102538024, 9.175232860164530, 9.609514739627184, 9.900159174019069, 10.29313547688341, 11.10412096337783, 11.49376993580503, 11.91569266145479, 12.47762284461594, 12.90150328179575, 13.31561883419425

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