Properties

Label 4-520e2-1.1-c1e2-0-6
Degree $4$
Conductor $270400$
Sign $1$
Analytic cond. $17.2409$
Root an. cond. $2.03769$
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  + 3·3-s − 2·5-s − 3·7-s + 3·9-s − 5·11-s − 2·13-s − 6·15-s − 3·17-s + 5·19-s − 9·21-s + 3·23-s + 3·25-s + 5·29-s + 16·31-s − 15·33-s + 6·35-s + 9·37-s − 6·39-s − 3·41-s + 43-s − 6·45-s − 24·47-s + 7·49-s − 9·51-s + 4·53-s + 10·55-s + 15·57-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.894·5-s − 1.13·7-s + 9-s − 1.50·11-s − 0.554·13-s − 1.54·15-s − 0.727·17-s + 1.14·19-s − 1.96·21-s + 0.625·23-s + 3/5·25-s + 0.928·29-s + 2.87·31-s − 2.61·33-s + 1.01·35-s + 1.47·37-s − 0.960·39-s − 0.468·41-s + 0.152·43-s − 0.894·45-s − 3.50·47-s + 49-s − 1.26·51-s + 0.549·53-s + 1.34·55-s + 1.98·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(270400\)    =    \(2^{6} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(17.2409\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 270400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.830213441\)
\(L(\frac12)\) \(\approx\) \(1.830213441\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
7$C_2^2$ \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - T - 88 T^{2} - p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−11.15773783095593600931307529550, −10.30307116726420003129175886689, −10.18449438595581796237140610088, −9.679953984654399956491786010887, −9.391744724807741738305258272243, −8.686318405517961695620533635450, −8.315698764866817697863712539006, −8.161226599088127856961603074011, −7.65812878364927031632105439058, −7.20104346398524019225269104052, −6.60836040231302759417409546644, −6.26902721788690842126075635335, −5.36376423050988807332413938945, −4.67760186966638720214880640810, −4.51991206602175920224873945799, −3.29854745563180318756575379034, −3.29211140995584102269353189495, −2.71583420946706239258448939980, −2.33609814881258665999937591805, −0.70949886755812806530210872213, 0.70949886755812806530210872213, 2.33609814881258665999937591805, 2.71583420946706239258448939980, 3.29211140995584102269353189495, 3.29854745563180318756575379034, 4.51991206602175920224873945799, 4.67760186966638720214880640810, 5.36376423050988807332413938945, 6.26902721788690842126075635335, 6.60836040231302759417409546644, 7.20104346398524019225269104052, 7.65812878364927031632105439058, 8.161226599088127856961603074011, 8.315698764866817697863712539006, 8.686318405517961695620533635450, 9.391744724807741738305258272243, 9.679953984654399956491786010887, 10.18449438595581796237140610088, 10.30307116726420003129175886689, 11.15773783095593600931307529550

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