Properties

Label 4-3360e2-1.1-c1e2-0-6
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  − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 4·13-s + 4·15-s + 4·17-s − 4·21-s + 8·23-s + 3·25-s − 4·27-s − 4·29-s − 4·35-s − 12·37-s + 8·39-s − 4·41-s + 8·43-s − 6·45-s + 3·49-s − 8·51-s + 4·53-s − 4·61-s + 6·63-s + 8·65-s + 8·67-s − 16·69-s + 16·71-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 1.10·13-s + 1.03·15-s + 0.970·17-s − 0.872·21-s + 1.66·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 0.676·35-s − 1.97·37-s + 1.28·39-s − 0.624·41-s + 1.21·43-s − 0.894·45-s + 3/7·49-s − 1.12·51-s + 0.549·53-s − 0.512·61-s + 0.755·63-s + 0.992·65-s + 0.977·67-s − 1.92·69-s + 1.89·71-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

\(L(1)\) \(\approx\) \(1.538053029\)
\(L(\frac12)\) \(\approx\) \(1.538053029\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 286 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−8.627423296709624906901707277084, −8.541178954689735725198196125960, −7.82768889697183784479006219341, −7.63546988022829457410782436562, −7.30421230268843493944950410527, −7.09701690393638375795132921434, −6.46401880863970492142727435641, −6.38769972009962741732693174729, −5.49161718975038720824069329861, −5.31713608872172631450254744367, −5.11654208665294952908115447561, −4.73463085838188686866199161074, −4.26315590316804096683565094468, −3.74956419366282075622876821452, −3.38765763728631859787280476714, −2.90266905089697740517576661562, −2.10296395468012365392438977092, −1.75658129212342272850500807411, −0.845784768645638452181556655321, −0.56468938323591597207244526122, 0.56468938323591597207244526122, 0.845784768645638452181556655321, 1.75658129212342272850500807411, 2.10296395468012365392438977092, 2.90266905089697740517576661562, 3.38765763728631859787280476714, 3.74956419366282075622876821452, 4.26315590316804096683565094468, 4.73463085838188686866199161074, 5.11654208665294952908115447561, 5.31713608872172631450254744367, 5.49161718975038720824069329861, 6.38769972009962741732693174729, 6.46401880863970492142727435641, 7.09701690393638375795132921434, 7.30421230268843493944950410527, 7.63546988022829457410782436562, 7.82768889697183784479006219341, 8.541178954689735725198196125960, 8.627423296709624906901707277084

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