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Properties

Label 2-369600-1.1-c1-0-126
Degree $2$
Conductor $369600$
Sign $-1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
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 − 6·13-s − 6·17-s + 21-s + 4·23-s − 27-s − 2·29-s + 33-s − 2·37-s + 6·39-s − 2·41-s − 8·43-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 4·59-s − 2·61-s − 63-s − 12·67-s − 4·69-s − 4·71-s + 6·73-s + 77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 1.45·17-s + 0.218·21-s + 0.834·23-s − 0.192·27-s − 0.371·29-s + 0.174·33-s − 0.328·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.520·59-s − 0.256·61-s − 0.125·63-s − 1.46·67-s − 0.481·69-s − 0.474·71-s + 0.702·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
11 \( 1 + T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.76690402968861, −12.24178462026238, −11.85307123511458, −11.37982907837245, −10.90359649257544, −10.56541994119043, −9.964099302876041, −9.626194915046058, −9.212570211230635, −8.692726613061797, −8.109694232749324, −7.593301984618159, −7.135646067422744, −6.657148124086520, −6.425889536067334, −5.713825892475637, −5.125676409382488, −4.740844113174128, −4.540299663317639, −3.658946937455971, −3.125783423486174, −2.610272183180476, −1.974688978704498, −1.509604767645060, −0.4452032657922060, 0, 0.4452032657922060, 1.509604767645060, 1.974688978704498, 2.610272183180476, 3.125783423486174, 3.658946937455971, 4.540299663317639, 4.740844113174128, 5.125676409382488, 5.713825892475637, 6.425889536067334, 6.657148124086520, 7.135646067422744, 7.593301984618159, 8.109694232749324, 8.692726613061797, 9.212570211230635, 9.626194915046058, 9.964099302876041, 10.56541994119043, 10.90359649257544, 11.37982907837245, 11.85307123511458, 12.24178462026238, 12.76690402968861

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