Properties

Label 4-2592e2-1.1-c1e2-0-15
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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  − 8·7-s + 10·17-s − 4·23-s + 6·25-s + 8·31-s − 10·41-s + 12·47-s + 34·49-s + 12·71-s + 18·73-s + 28·79-s − 28·89-s + 2·97-s + 12·103-s + 12·113-s − 80·119-s + 13·121-s + ⋯
L(s)  = 1  − 3.02·7-s + 2.42·17-s − 0.834·23-s + 6/5·25-s + 1.43·31-s − 1.56·41-s + 1.75·47-s + 34/7·49-s + 1.42·71-s + 2.10·73-s + 3.15·79-s − 2.96·89-s + 0.203·97-s + 1.18·103-s + 1.12·113-s − 7.33·119-s + 1.18·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.807425272\)
\(L(\frac12)\) \(\approx\) \(1.807425272\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.11.a_an
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
19$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.19.a_abl
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \) 2.43.a_bj
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \) 2.59.a_aen
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \) 2.67.a_aev
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.97.ac_hn
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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

−9.056790039904046581443592568504, −8.877039115066084812017234753564, −8.130698351421601421988342837329, −8.114126714555162090223280194443, −7.42835528083172887580093127716, −7.12344649649113331973845811093, −6.58081543741336949721027823851, −6.49707332372177201036679863871, −6.11434039723828817737245651052, −5.64813379326229438411499386793, −5.27899531674642223181679936990, −4.87014942433607626347001482681, −3.95713403100406123580054285921, −3.83937629775882937090934648874, −3.20857546817459366848380908624, −3.12776852230276948505832194140, −2.65146298955458765661610940902, −1.94371899044137482646927338356, −0.855496367201872642241971048822, −0.61110199134727875348818370995, 0.61110199134727875348818370995, 0.855496367201872642241971048822, 1.94371899044137482646927338356, 2.65146298955458765661610940902, 3.12776852230276948505832194140, 3.20857546817459366848380908624, 3.83937629775882937090934648874, 3.95713403100406123580054285921, 4.87014942433607626347001482681, 5.27899531674642223181679936990, 5.64813379326229438411499386793, 6.11434039723828817737245651052, 6.49707332372177201036679863871, 6.58081543741336949721027823851, 7.12344649649113331973845811093, 7.42835528083172887580093127716, 8.114126714555162090223280194443, 8.130698351421601421988342837329, 8.877039115066084812017234753564, 9.056790039904046581443592568504

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