Properties

Label 4-3360e2-1.1-c1e2-0-11
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 + 4·11-s − 4·15-s + 4·17-s + 4·19-s − 4·21-s + 3·25-s + 4·27-s − 4·29-s − 4·31-s + 8·33-s + 4·35-s + 4·37-s + 4·41-s + 8·43-s − 6·45-s + 8·47-s + 3·49-s + 8·51-s − 8·53-s − 8·55-s + 8·57-s + 4·61-s − 6·63-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s + 1.20·11-s − 1.03·15-s + 0.970·17-s + 0.917·19-s − 0.872·21-s + 3/5·25-s + 0.769·27-s − 0.742·29-s − 0.718·31-s + 1.39·33-s + 0.676·35-s + 0.657·37-s + 0.624·41-s + 1.21·43-s − 0.894·45-s + 1.16·47-s + 3/7·49-s + 1.12·51-s − 1.09·53-s − 1.07·55-s + 1.05·57-s + 0.512·61-s − 0.755·63-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.485406959\)
\(L(\frac12)\) \(\approx\) \(4.485406959\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 - 16 T + 182 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 32 T + 438 T^{2} - 32 p T^{3} + p^{2} T^{4} \)
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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

−8.769265212282106275604676518514, −8.576183215193029260027406693651, −7.83218974640952313532649440672, −7.78755421768061751153489709550, −7.32061362718310246475164860877, −7.27613768252847170711612820133, −6.56331729683429693267785476748, −6.33235717882000149368170322322, −5.80751418458941375949304291698, −5.47764583131319730686590448427, −4.74425290319223894922779945623, −4.54899778570738069265596064240, −3.84937731412763026995148717490, −3.60597315295901231959789155980, −3.45950003571848085079633022356, −2.98317713848799588506066931662, −2.27445908837267390691510594781, −1.95995435734588761818177844585, −0.965612699922309252692172287047, −0.76988563720808674405106281587, 0.76988563720808674405106281587, 0.965612699922309252692172287047, 1.95995435734588761818177844585, 2.27445908837267390691510594781, 2.98317713848799588506066931662, 3.45950003571848085079633022356, 3.60597315295901231959789155980, 3.84937731412763026995148717490, 4.54899778570738069265596064240, 4.74425290319223894922779945623, 5.47764583131319730686590448427, 5.80751418458941375949304291698, 6.33235717882000149368170322322, 6.56331729683429693267785476748, 7.27613768252847170711612820133, 7.32061362718310246475164860877, 7.78755421768061751153489709550, 7.83218974640952313532649440672, 8.576183215193029260027406693651, 8.769265212282106275604676518514

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