Properties

Label 2-369600-1.1-c1-0-142
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 − 6·13-s − 6·17-s + 2·19-s + 21-s + 6·23-s − 27-s − 2·29-s − 6·31-s + 33-s + 4·37-s + 6·39-s − 4·41-s + 12·43-s + 12·47-s + 49-s + 6·51-s + 10·53-s − 2·57-s + 4·59-s + 2·61-s − 63-s + 4·67-s − 6·69-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 + 0.458·19-s + 0.218·21-s + 1.25·23-s − 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.174·33-s + 0.657·37-s + 0.960·39-s − 0.624·41-s + 1.82·43-s + 1.75·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s − 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.125·63-s + 0.488·67-s − 0.722·69-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

\(L(1)\) \(\approx\) \(1.653944589\)
\(L(\frac12)\) \(\approx\) \(1.653944589\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.52618664727285, −12.10994550937221, −11.51128927083012, −11.19025285785883, −10.73612200197250, −10.29233907360886, −9.800094229861112, −9.337466860704941, −8.952577148022736, −8.567639163095051, −7.624563616463581, −7.383744393898126, −7.034956095203057, −6.605915702982348, −5.871322689012150, −5.485244764600889, −5.086682809239897, −4.479214906304524, −4.146691109549467, −3.454885404760819, −2.702155397000907, −2.382628794036468, −1.848032119181547, −0.7742100158454484, −0.4662426293048429, 0.4662426293048429, 0.7742100158454484, 1.848032119181547, 2.382628794036468, 2.702155397000907, 3.454885404760819, 4.146691109549467, 4.479214906304524, 5.086682809239897, 5.485244764600889, 5.871322689012150, 6.605915702982348, 7.034956095203057, 7.383744393898126, 7.624563616463581, 8.567639163095051, 8.952577148022736, 9.337466860704941, 9.800094229861112, 10.29233907360886, 10.73612200197250, 11.19025285785883, 11.51128927083012, 12.10994550937221, 12.52618664727285

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