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Properties

Label 2-369600-1.1-c1-0-152
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 + 7·17-s − 5·19-s − 21-s − 23-s − 27-s + 5·29-s + 8·31-s − 33-s − 2·37-s + 6·39-s + 12·41-s + 11·43-s + 8·47-s + 49-s − 7·51-s − 11·53-s + 5·57-s − 5·59-s − 7·61-s + 63-s + 2·67-s + 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 + 1.69·17-s − 1.14·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.928·29-s + 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.960·39-s + 1.87·41-s + 1.67·43-s + 1.16·47-s + 1/7·49-s − 0.980·51-s − 1.51·53-s + 0.662·57-s − 0.650·59-s − 0.896·61-s + 0.125·63-s + 0.244·67-s + 0.120·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.302435321\)
\(L(\frac12)\) \(\approx\) \(2.302435321\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.34522336915525, −12.08475594139078, −11.86742059956076, −10.99630218775020, −10.76152031878162, −10.22853761265281, −9.929933433551686, −9.312759623375021, −9.049927071005689, −8.242210262692572, −7.794610042243566, −7.543530925093842, −7.038437241585087, −6.330692596727116, −6.020481317817863, −5.580522029523549, −4.868906238036134, −4.553831099767422, −4.218895274679245, −3.431851460147054, −2.715788965009443, −2.435656965108730, −1.639707203953507, −0.9753282297249642, −0.4780846347095902, 0.4780846347095902, 0.9753282297249642, 1.639707203953507, 2.435656965108730, 2.715788965009443, 3.431851460147054, 4.218895274679245, 4.553831099767422, 4.868906238036134, 5.580522029523549, 6.020481317817863, 6.330692596727116, 7.038437241585087, 7.543530925093842, 7.794610042243566, 8.242210262692572, 9.049927071005689, 9.312759623375021, 9.929933433551686, 10.22853761265281, 10.76152031878162, 10.99630218775020, 11.86742059956076, 12.08475594139078, 12.34522336915525

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