Properties

Label 4-442368-1.1-c1e2-0-20
Degree $4$
Conductor $442368$
Sign $1$
Analytic cond. $28.2057$
Root an. cond. $2.30454$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s + 8·13-s + 8·23-s − 2·25-s + 27-s − 16·37-s + 8·39-s − 8·47-s − 2·49-s + 24·59-s + 8·69-s + 8·71-s + 12·73-s − 2·75-s + 81-s + 16·83-s − 12·97-s + 8·107-s − 8·109-s − 16·111-s + 8·117-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 2.21·13-s + 1.66·23-s − 2/5·25-s + 0.192·27-s − 2.63·37-s + 1.28·39-s − 1.16·47-s − 2/7·49-s + 3.12·59-s + 0.963·69-s + 0.949·71-s + 1.40·73-s − 0.230·75-s + 1/9·81-s + 1.75·83-s − 1.21·97-s + 0.773·107-s − 0.766·109-s − 1.51·111-s + 0.739·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(442368\)    =    \(2^{14} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(28.2057\)
Root analytic conductor: \(2.30454\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{442368} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 442368,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.751738390\)
\(L(\frac12)\) \(\approx\) \(2.751738390\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
29$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.675158491675165215369551746147, −8.218013526102369441905502243738, −7.86704922425443562972388219098, −7.10748790571745697898758050070, −6.66176083419083012318294992252, −6.56892658471217228215068116190, −5.70250558287385901640498669995, −5.29180031661113696569524704419, −4.85209493895782004684614169946, −3.99706689441056134881647785711, −3.50084386192981639636412108046, −3.38214495741844609030455918031, −2.40127302667982511847456716398, −1.67459054552091998302273658202, −0.947975153191257896567865420144, 0.947975153191257896567865420144, 1.67459054552091998302273658202, 2.40127302667982511847456716398, 3.38214495741844609030455918031, 3.50084386192981639636412108046, 3.99706689441056134881647785711, 4.85209493895782004684614169946, 5.29180031661113696569524704419, 5.70250558287385901640498669995, 6.56892658471217228215068116190, 6.66176083419083012318294992252, 7.10748790571745697898758050070, 7.86704922425443562972388219098, 8.218013526102369441905502243738, 8.675158491675165215369551746147

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