Properties

Label 4-336e2-1.1-c1e2-0-36
Degree $4$
Conductor $112896$
Sign $1$
Analytic cond. $7.19834$
Root an. cond. $1.63797$
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 − 7-s + 6·9-s + 9·19-s − 3·21-s + 5·25-s + 9·27-s − 15·31-s − 37-s + 10·43-s − 6·49-s + 27·57-s + 12·61-s − 6·63-s + 11·67-s − 27·73-s + 15·75-s − 13·79-s + 9·81-s − 45·93-s − 33·103-s + 17·109-s − 3·111-s − 11·121-s + 127-s + 30·129-s + 131-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.377·7-s + 2·9-s + 2.06·19-s − 0.654·21-s + 25-s + 1.73·27-s − 2.69·31-s − 0.164·37-s + 1.52·43-s − 6/7·49-s + 3.57·57-s + 1.53·61-s − 0.755·63-s + 1.34·67-s − 3.16·73-s + 1.73·75-s − 1.46·79-s + 81-s − 4.66·93-s − 3.25·103-s + 1.62·109-s − 0.284·111-s − 121-s + 0.0887·127-s + 2.64·129-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(112896\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(7.19834\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 112896,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.931350310\)
\(L(\frac12)\) \(\approx\) \(2.931350310\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \)
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−11.67660316037631171716828806208, −11.41941785603222522887957765342, −10.69686699709282516646201134262, −10.33981794205294891009798878631, −9.533009708702607669203660333621, −9.531179361467725547520808615724, −9.089326327690703632501222123323, −8.525240011691245440657276104083, −8.107002892703647091153826134274, −7.49347509077663509375304093656, −7.04881314349125167750816694374, −6.96992794580905081562518928567, −5.69065440049857888425647511780, −5.53399340368707844810123254047, −4.59530871416190169385855182024, −3.97616432998908503075031220488, −3.29200566794896011726735625410, −3.04444386945885656669642568983, −2.18030472597969985171433190390, −1.29473696278255747630958387854, 1.29473696278255747630958387854, 2.18030472597969985171433190390, 3.04444386945885656669642568983, 3.29200566794896011726735625410, 3.97616432998908503075031220488, 4.59530871416190169385855182024, 5.53399340368707844810123254047, 5.69065440049857888425647511780, 6.96992794580905081562518928567, 7.04881314349125167750816694374, 7.49347509077663509375304093656, 8.107002892703647091153826134274, 8.525240011691245440657276104083, 9.089326327690703632501222123323, 9.531179361467725547520808615724, 9.533009708702607669203660333621, 10.33981794205294891009798878631, 10.69686699709282516646201134262, 11.41941785603222522887957765342, 11.67660316037631171716828806208

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