Properties

Label 4-48e4-1.1-c2e2-0-7
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $3941.25$
Root an. cond. $7.92334$
Motivic weight $2$
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  + 4·5-s − 38·25-s − 100·29-s + 2·49-s + 188·53-s − 100·73-s − 380·97-s + 380·101-s − 142·121-s − 268·125-s + 127-s + 131-s + 137-s + 139-s − 400·145-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 4/5·5-s − 1.51·25-s − 3.44·29-s + 2/49·49-s + 3.54·53-s − 1.36·73-s − 3.91·97-s + 3.76·101-s − 1.17·121-s − 2.14·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.75·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(3941.25\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2304} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.240458850\)
\(L(\frac12)\) \(\approx\) \(1.240458850\)
\(L(2)\) 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 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
7$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \)
11$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
19$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
29$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 38 T + p^{2} T^{2} )( 1 + 38 T + p^{2} T^{2} ) \)
37$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
53$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2$ \( ( 1 + 50 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 58 T + p^{2} T^{2} )( 1 + 58 T + p^{2} T^{2} ) \)
83$C_2$ \( ( 1 - 134 T + p^{2} T^{2} )( 1 + 134 T + p^{2} T^{2} ) \)
89$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
97$C_2$ \( ( 1 + 190 T + p^{2} T^{2} )^{2} \)
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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

−9.252323473804554270994662583211, −8.603258325580597282536607305726, −8.379484714569256146536086506080, −7.65795639840418766304199506872, −7.59664215659282173926497508411, −6.97379956533045215473993456100, −6.93984244779167864714567609113, −5.97133742641685569718494832400, −5.92937407018795759817014023577, −5.54040420567272753428225089478, −5.32174954055330780145144605827, −4.57252794877238667427058945948, −4.09818071117764543335843482563, −3.72107313182437970526660117334, −3.41425095394894885275083059965, −2.52211840332669157915131240029, −2.21548733895160069777541257204, −1.78068829062375822885783506935, −1.19926428104994108417954238602, −0.26688735975999893160163739402, 0.26688735975999893160163739402, 1.19926428104994108417954238602, 1.78068829062375822885783506935, 2.21548733895160069777541257204, 2.52211840332669157915131240029, 3.41425095394894885275083059965, 3.72107313182437970526660117334, 4.09818071117764543335843482563, 4.57252794877238667427058945948, 5.32174954055330780145144605827, 5.54040420567272753428225089478, 5.92937407018795759817014023577, 5.97133742641685569718494832400, 6.93984244779167864714567609113, 6.97379956533045215473993456100, 7.59664215659282173926497508411, 7.65795639840418766304199506872, 8.379484714569256146536086506080, 8.603258325580597282536607305726, 9.252323473804554270994662583211

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