Properties

Label 4-520e2-1.1-c1e2-0-25
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-s + 2·5-s + 3·7-s + 3·9-s − 5·11-s + 2·13-s + 2·15-s − 5·17-s + 19-s + 3·21-s + 23-s + 3·25-s + 8·27-s + 9·29-s − 8·31-s − 5·33-s + 6·35-s + 3·37-s + 2·39-s + 41-s + 3·43-s + 6·45-s − 16·47-s + 7·49-s − 5·51-s + 20·53-s − 10·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1.13·7-s + 9-s − 1.50·11-s + 0.554·13-s + 0.516·15-s − 1.21·17-s + 0.229·19-s + 0.654·21-s + 0.208·23-s + 3/5·25-s + 1.53·27-s + 1.67·29-s − 1.43·31-s − 0.870·33-s + 1.01·35-s + 0.493·37-s + 0.320·39-s + 0.156·41-s + 0.457·43-s + 0.894·45-s − 2.33·47-s + 49-s − 0.700·51-s + 2.74·53-s − 1.34·55-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

\(L(1)\) \(\approx\) \(3.046362406\)
\(L(\frac12)\) \(\approx\) \(3.046362406\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
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$C_2^2$ \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 7 T - 22 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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

−11.01542403512380006002016574327, −10.52932512010179000266898876458, −10.25124459112204285904229883630, −9.923263481606343068786950881748, −9.193066708833822758866470009660, −8.882726585899490539485896535861, −8.324946355564042927877110080429, −8.204842278375652234735248115935, −7.39105662258052018084773668426, −7.21894418167182659891199126203, −6.48767775415952170940638627246, −6.11500117052107891339853449249, −5.19896586631692286332346362635, −5.10625409102093016625191997986, −4.51329524636558166563095222785, −3.93700498026246250661971904892, −3.01865658289825316781432794654, −2.45764912152010748406382100469, −1.93573540966776501683835072665, −1.10491614064019680049233612752, 1.10491614064019680049233612752, 1.93573540966776501683835072665, 2.45764912152010748406382100469, 3.01865658289825316781432794654, 3.93700498026246250661971904892, 4.51329524636558166563095222785, 5.10625409102093016625191997986, 5.19896586631692286332346362635, 6.11500117052107891339853449249, 6.48767775415952170940638627246, 7.21894418167182659891199126203, 7.39105662258052018084773668426, 8.204842278375652234735248115935, 8.324946355564042927877110080429, 8.882726585899490539485896535861, 9.193066708833822758866470009660, 9.923263481606343068786950881748, 10.25124459112204285904229883630, 10.52932512010179000266898876458, 11.01542403512380006002016574327

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