Properties

Label 2-369600-1.1-c1-0-114
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 11-s − 4·13-s + 21-s + 6·23-s − 27-s + 2·29-s + 33-s + 4·37-s + 4·39-s − 6·41-s + 4·43-s − 4·47-s + 49-s + 6·53-s − 10·61-s − 63-s + 10·67-s − 6·69-s + 8·71-s − 14·73-s + 77-s − 8·79-s + 81-s − 2·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.218·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s + 0.174·33-s + 0.657·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s − 0.125·63-s + 1.22·67-s − 0.722·69-s + 0.949·71-s − 1.63·73-s + 0.113·77-s − 0.900·79-s + 1/9·81-s − 0.214·87-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: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.621824532\)
\(L(\frac12)\) \(\approx\) \(1.621824532\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−12.47583556243610, −12.11758495674399, −11.53066678838305, −11.26797460597607, −10.55140091662213, −10.39977552770429, −9.745115253144104, −9.472833760943494, −8.942650465398086, −8.372253784658722, −7.879158013787635, −7.331031814635309, −6.895673056701369, −6.662087195620695, −5.795088800135959, −5.650744285421622, −4.914905043296902, −4.624462947167888, −4.131861500822421, −3.208129405121310, −3.045752763734525, −2.282528410819371, −1.756531046809623, −0.8860161786615790, −0.4194817571811204, 0.4194817571811204, 0.8860161786615790, 1.756531046809623, 2.282528410819371, 3.045752763734525, 3.208129405121310, 4.131861500822421, 4.624462947167888, 4.914905043296902, 5.650744285421622, 5.795088800135959, 6.662087195620695, 6.895673056701369, 7.331031814635309, 7.879158013787635, 8.372253784658722, 8.942650465398086, 9.472833760943494, 9.745115253144104, 10.39977552770429, 10.55140091662213, 11.26797460597607, 11.53066678838305, 12.11758495674399, 12.47583556243610

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