Properties

Label 2-369600-1.1-c1-0-137
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 + 2·17-s + 2·19-s + 21-s + 27-s + 4·29-s + 8·31-s − 33-s − 2·37-s − 6·39-s − 4·41-s + 8·43-s − 6·47-s + 49-s + 2·51-s + 2·53-s + 2·57-s − 4·59-s + 6·61-s + 63-s + 10·67-s − 4·71-s − 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.485·17-s + 0.458·19-s + 0.218·21-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.960·39-s − 0.624·41-s + 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.264·57-s − 0.520·59-s + 0.768·61-s + 0.125·63-s + 1.22·67-s − 0.474·71-s − 1.17·73-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

\(L(1)\) \(\approx\) \(3.122696940\)
\(L(\frac12)\) \(\approx\) \(3.122696940\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.48618163601357, −12.06605665702067, −11.69790223225208, −11.23261913678357, −10.53304982889369, −10.10277745795646, −9.846528177818483, −9.462647970703877, −8.686657303307912, −8.486331107436160, −7.826904853860295, −7.584424293808803, −7.052694723775476, −6.649348795033944, −5.961218521641367, −5.361902966250574, −4.991901776148166, −4.463127814763796, −4.101874155333005, −3.175108951629648, −2.933433555729104, −2.379874267272237, −1.830335686087542, −1.114300441984336, −0.4580433226466511, 0.4580433226466511, 1.114300441984336, 1.830335686087542, 2.379874267272237, 2.933433555729104, 3.175108951629648, 4.101874155333005, 4.463127814763796, 4.991901776148166, 5.361902966250574, 5.961218521641367, 6.649348795033944, 7.052694723775476, 7.584424293808803, 7.826904853860295, 8.486331107436160, 8.686657303307912, 9.462647970703877, 9.846528177818483, 10.10277745795646, 10.53304982889369, 11.23261913678357, 11.69790223225208, 12.06605665702067, 12.48618163601357

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