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Properties

Label 2-369600-1.1-c1-0-112
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 + 2·13-s − 6·17-s + 21-s + 27-s + 2·29-s + 33-s − 2·37-s + 2·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s − 6·51-s − 6·53-s − 10·61-s + 63-s + 8·67-s + 16·71-s + 10·73-s + 77-s + 8·79-s + 81-s + 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 1.45·17-s + 0.218·21-s + 0.192·27-s + 0.371·29-s + 0.174·33-s − 0.328·37-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 − 1.28·61-s + 0.125·63-s + 0.977·67-s + 1.89·71-s + 1.17·73-s + 0.113·77-s + 0.900·79-s + 1/9·81-s + 1.31·83-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: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.881832317\)
\(L(\frac12)\) \(\approx\) \(2.881832317\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.52863417428177, −12.10832259122711, −11.47463980577802, −11.17702861393212, −10.73977984560787, −10.28962451279730, −9.599726720541309, −9.375731189441803, −8.784789735021126, −8.403867750594901, −8.055499674639000, −7.567784712116095, −6.834560307890253, −6.518973299798072, −6.270134958288567, −5.315624703074099, −4.964677056663056, −4.501192898043752, −3.872458825451123, −3.480971548870263, −2.926002865131110, −2.167417634555289, −1.863496321945578, −1.212631058815739, −0.4181052541213343, 0.4181052541213343, 1.212631058815739, 1.863496321945578, 2.167417634555289, 2.926002865131110, 3.480971548870263, 3.872458825451123, 4.501192898043752, 4.964677056663056, 5.315624703074099, 6.270134958288567, 6.518973299798072, 6.834560307890253, 7.567784712116095, 8.055499674639000, 8.403867750594901, 8.784789735021126, 9.375731189441803, 9.599726720541309, 10.28962451279730, 10.73977984560787, 11.17702861393212, 11.47463980577802, 12.10832259122711, 12.52863417428177

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