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Properties

Label 2-193200-1.1-c1-0-141
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 + 4·11-s + 2·13-s + 6·17-s − 4·19-s − 21-s − 23-s − 27-s − 2·29-s + 8·31-s − 4·33-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s − 6·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s + 63-s + 4·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s + 0.488·67-s + 0.120·69-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.41975631399063, −12.54120472208911, −12.47482348692191, −11.87772723920868, −11.45050886648925, −11.15568350642280, −10.46017333215188, −10.05836274175660, −9.725909229503009, −8.967498291752861, −8.654545107347264, −7.922550736278188, −7.807313989861393, −6.758460741777344, −6.665254489299322, −6.100627685757716, −5.595807250808585, −4.948915656108230, −4.587241465767570, −3.861944532759891, −3.493598719903787, −2.875774515274565, −1.834140833231806, −1.520214850047229, −0.8959067829622875, 0, 0.8959067829622875, 1.520214850047229, 1.834140833231806, 2.875774515274565, 3.493598719903787, 3.861944532759891, 4.587241465767570, 4.948915656108230, 5.595807250808585, 6.100627685757716, 6.665254489299322, 6.758460741777344, 7.807313989861393, 7.922550736278188, 8.654545107347264, 8.967498291752861, 9.725909229503009, 10.05836274175660, 10.46017333215188, 11.15568350642280, 11.45050886648925, 11.87772723920868, 12.47482348692191, 12.54120472208911, 13.41975631399063

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