Properties

Label 4-1216e2-1.1-c1e2-0-15
Degree $4$
Conductor $1478656$
Sign $1$
Analytic cond. $94.2803$
Root an. cond. $3.11605$
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  + 2·5-s − 2·9-s + 8·13-s + 10·17-s − 7·25-s − 4·29-s + 20·37-s + 12·41-s − 4·45-s − 5·49-s + 16·53-s + 10·61-s + 16·65-s − 30·73-s − 5·81-s + 20·85-s + 32·97-s + 36·101-s − 24·109-s + 4·113-s − 16·117-s − 13·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.894·5-s − 2/3·9-s + 2.21·13-s + 2.42·17-s − 7/5·25-s − 0.742·29-s + 3.28·37-s + 1.87·41-s − 0.596·45-s − 5/7·49-s + 2.19·53-s + 1.28·61-s + 1.98·65-s − 3.51·73-s − 5/9·81-s + 2.16·85-s + 3.24·97-s + 3.58·101-s − 2.29·109-s + 0.376·113-s − 1.47·117-s − 1.18·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1478656\)    =    \(2^{12} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(94.2803\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1478656} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1478656,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.397222379\)
\(L(\frac12)\) \(\approx\) \(3.397222379\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
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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

−8.024247643273632491999690471751, −7.43537131861627444263317650332, −7.30495770738943457952049118581, −6.20879084669067007942487517486, −5.95681555667055397919593180547, −5.89904763388088518363904025264, −5.67735294850741520947496000769, −4.88105113826037271671762872708, −4.19084046620099313866498183608, −3.73244909832136496864835491552, −3.41145508666670040670114085640, −2.67622771685970545269516699451, −2.20146006630293663601866411040, −1.26674131115266326801058270565, −0.944005207983174855702403672277, 0.944005207983174855702403672277, 1.26674131115266326801058270565, 2.20146006630293663601866411040, 2.67622771685970545269516699451, 3.41145508666670040670114085640, 3.73244909832136496864835491552, 4.19084046620099313866498183608, 4.88105113826037271671762872708, 5.67735294850741520947496000769, 5.89904763388088518363904025264, 5.95681555667055397919593180547, 6.20879084669067007942487517486, 7.30495770738943457952049118581, 7.43537131861627444263317650332, 8.024247643273632491999690471751

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