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Properties

Label 2-369600-1.1-c1-0-109
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 + 5·13-s − 3·17-s − 7·19-s + 21-s + 8·23-s − 27-s + 6·29-s − 33-s + 3·37-s − 5·39-s − 3·41-s − 4·43-s + 4·47-s + 49-s + 3·51-s + 7·53-s + 7·57-s + 2·59-s − 61-s − 63-s + 3·67-s − 8·69-s − 15·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.38·13-s − 0.727·17-s − 1.60·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s + 1.11·29-s − 0.174·33-s + 0.493·37-s − 0.800·39-s − 0.468·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.420·51-s + 0.961·53-s + 0.927·57-s + 0.260·59-s − 0.128·61-s − 0.125·63-s + 0.366·67-s − 0.963·69-s − 1.78·71-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

\(L(1)\) \(\approx\) \(1.845846025\)
\(L(\frac12)\) \(\approx\) \(1.845846025\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.49401140497594, −12.07908964168784, −11.47482711304228, −11.14024686455452, −10.72328137561537, −10.42087381298953, −9.872293685235356, −9.255797830402605, −8.741190440537879, −8.568916661967974, −8.084881677467516, −7.150300795623568, −6.920845428907493, −6.497220388092665, −6.005959105433475, −5.701017498245713, −4.858031940538154, −4.565455770612444, −3.998002257173295, −3.538391409618381, −2.860454203732352, −2.350534119268435, −1.567566132672232, −1.057567651524755, −0.4091351749931699, 0.4091351749931699, 1.057567651524755, 1.567566132672232, 2.350534119268435, 2.860454203732352, 3.538391409618381, 3.998002257173295, 4.565455770612444, 4.858031940538154, 5.701017498245713, 6.005959105433475, 6.497220388092665, 6.920845428907493, 7.150300795623568, 8.084881677467516, 8.568916661967974, 8.741190440537879, 9.255797830402605, 9.872293685235356, 10.42087381298953, 10.72328137561537, 11.14024686455452, 11.47482711304228, 12.07908964168784, 12.49401140497594

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