Properties

Label 4-4400e2-1.1-c1e2-0-21
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5·7-s + 2·11-s − 2·13-s − 3·17-s − 4·19-s + 5·21-s + 3·23-s + 2·27-s + 9·29-s + 8·31-s + 2·33-s + 4·37-s − 2·39-s − 12·41-s + 20·43-s + 12·47-s + 10·49-s − 3·51-s + 9·53-s − 4·57-s + 12·59-s + 13·61-s + 8·67-s + 3·69-s − 6·71-s − 11·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.88·7-s + 0.603·11-s − 0.554·13-s − 0.727·17-s − 0.917·19-s + 1.09·21-s + 0.625·23-s + 0.384·27-s + 1.67·29-s + 1.43·31-s + 0.348·33-s + 0.657·37-s − 0.320·39-s − 1.87·41-s + 3.04·43-s + 1.75·47-s + 10/7·49-s − 0.420·51-s + 1.23·53-s − 0.529·57-s + 1.56·59-s + 1.66·61-s + 0.977·67-s + 0.361·69-s − 0.712·71-s − 1.28·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.862897725\)
\(L(\frac12)\) \(\approx\) \(5.862897725\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 9 T + 73 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 9 T + 79 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 133 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 6 T - 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + T - 99 T^{2} + p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 3 T + 37 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 3 T + 175 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 17 T + 261 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
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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

−8.456488576740967865823679941994, −8.328405098369581229062770170256, −7.959797382753944072572024644275, −7.51405040935856584931939751583, −6.97412920141792918552110114479, −6.92465015189509998346135590936, −6.50310068300794013093100224175, −5.93250856137654351978308251185, −5.46751052839221346073436886476, −5.25114435780144175260731942785, −4.63565067590259918373434520755, −4.33727799683710847669317652175, −4.23955976748951590746915802473, −3.76864890588120928453687305295, −2.76135413042073571643368970826, −2.72559048907944102315234968023, −2.29438917904943086374888215499, −1.76098752075655169183009455484, −1.05728421614977377908413900746, −0.76771904926768841856917877706, 0.76771904926768841856917877706, 1.05728421614977377908413900746, 1.76098752075655169183009455484, 2.29438917904943086374888215499, 2.72559048907944102315234968023, 2.76135413042073571643368970826, 3.76864890588120928453687305295, 4.23955976748951590746915802473, 4.33727799683710847669317652175, 4.63565067590259918373434520755, 5.25114435780144175260731942785, 5.46751052839221346073436886476, 5.93250856137654351978308251185, 6.50310068300794013093100224175, 6.92465015189509998346135590936, 6.97412920141792918552110114479, 7.51405040935856584931939751583, 7.959797382753944072572024644275, 8.328405098369581229062770170256, 8.456488576740967865823679941994

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