Properties

Label 2-193200-1.1-c1-0-18
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 5·11-s − 2·13-s + 2·17-s + 19-s + 21-s − 23-s + 27-s − 4·29-s − 5·33-s + 6·37-s − 2·39-s − 3·41-s − 2·43-s − 13·47-s + 49-s + 2·51-s + 3·53-s + 57-s + 4·59-s − 10·61-s + 63-s + 14·67-s − 69-s − 14·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.742·29-s − 0.870·33-s + 0.986·37-s − 0.320·39-s − 0.468·41-s − 0.304·43-s − 1.89·47-s + 1/7·49-s + 0.280·51-s + 0.412·53-s + 0.132·57-s + 0.520·59-s − 1.28·61-s + 0.125·63-s + 1.71·67-s − 0.120·69-s − 1.66·71-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: $\chi_{193200} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 193200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.611882872\)
\(L(\frac12)\) \(\approx\) \(1.611882872\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−13.11973617876575, −12.69651726430484, −12.28632243443163, −11.63428025903160, −11.22195481531518, −10.71911703328967, −10.18687889110864, −9.724606221953718, −9.512503387189424, −8.662969730051289, −8.280130857882182, −7.815901400051948, −7.551379273601655, −6.949336950934670, −6.381537203333721, −5.576031094498715, −5.328045963726301, −4.753748959121431, −4.232246198816240, −3.528361083141330, −2.970885912420723, −2.534321222853600, −1.913620565978435, −1.305917676347670, −0.3356606613659696, 0.3356606613659696, 1.305917676347670, 1.913620565978435, 2.534321222853600, 2.970885912420723, 3.528361083141330, 4.232246198816240, 4.753748959121431, 5.328045963726301, 5.576031094498715, 6.381537203333721, 6.949336950934670, 7.551379273601655, 7.815901400051948, 8.280130857882182, 8.662969730051289, 9.512503387189424, 9.724606221953718, 10.18687889110864, 10.71911703328967, 11.22195481531518, 11.63428025903160, 12.28632243443163, 12.69651726430484, 13.11973617876575

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