Properties

Label 4-3360e2-1.1-c1e2-0-12
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 9-s + 12·11-s − 4·19-s − 25-s + 20·29-s + 4·31-s − 12·41-s − 2·45-s − 49-s + 24·55-s − 8·59-s − 4·61-s + 20·71-s + 16·79-s + 81-s − 12·89-s − 8·95-s − 12·99-s − 12·101-s + 28·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 1/3·9-s + 3.61·11-s − 0.917·19-s − 1/5·25-s + 3.71·29-s + 0.718·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s + 3.23·55-s − 1.04·59-s − 0.512·61-s + 2.37·71-s + 1.80·79-s + 1/9·81-s − 1.27·89-s − 0.820·95-s − 1.20·99-s − 1.19·101-s + 2.68·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.068937468\)
\(L(\frac12)\) \(\approx\) \(5.068937468\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.796020192331507119636215240498, −8.466870426476450008997059888400, −8.289911746358335331627278554750, −7.82994010967190328918995817410, −6.93512531210345203240943310681, −6.84135048254287587183769791616, −6.48988068066323899172472099432, −6.22292210172369771422616926911, −6.18040173936804991605583051804, −5.41678182619257942012005721138, −4.86105972778239050161160368367, −4.60270304196198113827566185855, −4.10019212402593717116952205368, −3.86004069943676364699251322876, −3.15456384005629271568839755965, −2.96347171776681042851145701708, −2.03933202382985221797193432272, −1.82565225667228421790940126462, −1.14365025470341681051492998032, −0.796839997129157072370572350958, 0.796839997129157072370572350958, 1.14365025470341681051492998032, 1.82565225667228421790940126462, 2.03933202382985221797193432272, 2.96347171776681042851145701708, 3.15456384005629271568839755965, 3.86004069943676364699251322876, 4.10019212402593717116952205368, 4.60270304196198113827566185855, 4.86105972778239050161160368367, 5.41678182619257942012005721138, 6.18040173936804991605583051804, 6.22292210172369771422616926911, 6.48988068066323899172472099432, 6.84135048254287587183769791616, 6.93512531210345203240943310681, 7.82994010967190328918995817410, 8.289911746358335331627278554750, 8.466870426476450008997059888400, 8.796020192331507119636215240498

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