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Properties

Label 4-48e4-1.1-c1e2-0-19
Degree $4$
Conductor $5308416$
Sign $1$
Analytic cond. $338.469$
Root an. cond. $4.28923$
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  + 4·7-s − 2·25-s + 20·31-s − 2·49-s − 28·73-s + 20·79-s + 4·97-s + 28·103-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s − 8·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  + 1.51·7-s − 2/5·25-s + 3.59·31-s − 2/7·49-s − 3.27·73-s + 2.25·79-s + 0.406·97-s + 2.75·103-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s − 0.604·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(5308416\)    =    \(2^{16} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(338.469\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5308416,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.325806782\)
\(L(\frac12)\) \(\approx\) \(3.325806782\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.29.a_by
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.53.a_adq
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.83.a_fe
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−9.136398946569171392790795517249, −8.618850349946029972549487438837, −8.361255339148681707197923814251, −8.190317218400559501156480400180, −7.65257603976594975823956538711, −7.44896606746174051922701192764, −6.91454146600431427713492292885, −6.36523513407901939637797751002, −6.13700566218195458354613235626, −5.72841593976055951052620493493, −4.99907292809383628661532444156, −4.88450018484767907656374024991, −4.30943750807639038667710226447, −4.29753523993744654552708635154, −3.20401742467794596651434910843, −3.14374695409377479797205561887, −2.30212776630714975831747505426, −1.92621689418655432325939293744, −1.26177245514393104920732673595, −0.68443941890410295313288140747, 0.68443941890410295313288140747, 1.26177245514393104920732673595, 1.92621689418655432325939293744, 2.30212776630714975831747505426, 3.14374695409377479797205561887, 3.20401742467794596651434910843, 4.29753523993744654552708635154, 4.30943750807639038667710226447, 4.88450018484767907656374024991, 4.99907292809383628661532444156, 5.72841593976055951052620493493, 6.13700566218195458354613235626, 6.36523513407901939637797751002, 6.91454146600431427713492292885, 7.44896606746174051922701192764, 7.65257603976594975823956538711, 8.190317218400559501156480400180, 8.361255339148681707197923814251, 8.618850349946029972549487438837, 9.136398946569171392790795517249

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