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Properties

Label 2-193200-1.1-c1-0-68
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 − 6·17-s − 7·19-s − 21-s − 23-s − 27-s − 6·29-s − 4·31-s − 33-s + 2·37-s + 4·39-s + 9·41-s + 2·43-s + 7·47-s + 49-s + 6·51-s − 5·53-s + 7·57-s − 7·59-s − 7·61-s + 63-s − 2·67-s + 69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 1.45·17-s − 1.60·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.640·39-s + 1.40·41-s + 0.304·43-s + 1.02·47-s + 1/7·49-s + 0.840·51-s − 0.686·53-s + 0.927·57-s − 0.911·59-s − 0.896·61-s + 0.125·63-s − 0.244·67-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 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
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

−13.27652917447021, −12.76396050324471, −12.31312042492560, −12.09968704285548, −11.23588140574389, −10.99179218458420, −10.75606288937173, −10.11965144632697, −9.428224817867636, −9.115879072786474, −8.758823496923563, −7.850332753675671, −7.668362832654133, −7.046713589560804, −6.493117509796419, −6.130364795997510, −5.577527443569171, −4.911594305992861, −4.458071578449467, −4.155020843724743, −3.476538354198080, −2.478277808122400, −2.181956692943037, −1.628569193294323, −0.6066097692491230, 0, 0.6066097692491230, 1.628569193294323, 2.181956692943037, 2.478277808122400, 3.476538354198080, 4.155020843724743, 4.458071578449467, 4.911594305992861, 5.577527443569171, 6.130364795997510, 6.493117509796419, 7.046713589560804, 7.668362832654133, 7.850332753675671, 8.758823496923563, 9.115879072786474, 9.428224817867636, 10.11965144632697, 10.75606288937173, 10.99179218458420, 11.23588140574389, 12.09968704285548, 12.31312042492560, 12.76396050324471, 13.27652917447021

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