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Properties

Label 2-369600-1.1-c1-0-390
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s + 9-s + 11-s − 4·13-s − 2·17-s − 21-s + 6·23-s + 27-s − 2·31-s + 33-s − 8·37-s − 4·39-s + 2·41-s + 4·43-s − 8·47-s + 49-s − 2·51-s + 6·53-s − 2·61-s − 63-s + 8·67-s + 6·69-s + 8·71-s + 4·73-s − 77-s − 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.218·21-s + 1.25·23-s + 0.192·27-s − 0.359·31-s + 0.174·33-s − 1.31·37-s − 0.640·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.722·69-s + 0.949·71-s + 0.468·73-s − 0.113·77-s − 1.12·79-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: \(1\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.76293454615181, −12.36323309579190, −11.90011224606013, −11.34681250977269, −10.91359320571994, −10.42600729760842, −9.922119406479188, −9.467240795966470, −9.212581392305770, −8.552814952308901, −8.369283961387137, −7.532817644971580, −7.262206209690244, −6.801312097939996, −6.438613751022035, −5.659155271017998, −5.215863984249680, −4.711868173741697, −4.236330766298379, −3.571668608305580, −3.199709341381934, −2.565303950917322, −2.151964713608597, −1.490944110578556, −0.7491298431572280, 0, 0.7491298431572280, 1.490944110578556, 2.151964713608597, 2.565303950917322, 3.199709341381934, 3.571668608305580, 4.236330766298379, 4.711868173741697, 5.215863984249680, 5.659155271017998, 6.438613751022035, 6.801312097939996, 7.262206209690244, 7.532817644971580, 8.369283961387137, 8.552814952308901, 9.212581392305770, 9.467240795966470, 9.922119406479188, 10.42600729760842, 10.91359320571994, 11.34681250977269, 11.90011224606013, 12.36323309579190, 12.76293454615181

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