Properties

Label 2-369600-1.1-c1-0-12
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 − 2·17-s − 4·19-s + 21-s + 8·23-s + 27-s − 6·29-s − 8·31-s − 33-s − 2·37-s − 2·39-s + 2·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 4·57-s − 12·59-s + 2·61-s + 63-s + 4·67-s + 8·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.963·69-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

\(L(1)\) \(\approx\) \(0.7456521515\)
\(L(\frac12)\) \(\approx\) \(0.7456521515\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−12.62335709498998, −12.24322285506693, −11.40994785511167, −11.03746089149233, −10.87783584617932, −10.24103884931862, −9.710092008893580, −9.214600285921232, −8.878749652305698, −8.516138437157267, −7.899687951041020, −7.363277529690508, −7.181149127490809, −6.618086569788780, −5.927759308634099, −5.493678265181483, −4.844464798246992, −4.601917810391340, −3.875034290090583, −3.497350969084350, −2.636299003934731, −2.548695674768766, −1.611125289869130, −1.397431116774041, −0.1989655507303734, 0.1989655507303734, 1.397431116774041, 1.611125289869130, 2.548695674768766, 2.636299003934731, 3.497350969084350, 3.875034290090583, 4.601917810391340, 4.844464798246992, 5.493678265181483, 5.927759308634099, 6.618086569788780, 7.181149127490809, 7.363277529690508, 7.899687951041020, 8.516138437157267, 8.878749652305698, 9.214600285921232, 9.710092008893580, 10.24103884931862, 10.87783584617932, 11.03746089149233, 11.40994785511167, 12.24322285506693, 12.62335709498998

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