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Properties

Label 4-3888e2-1.1-c1e2-0-6
Degree $4$
Conductor $15116544$
Sign $1$
Analytic cond. $963.843$
Root an. cond. $5.57187$
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  − 10·13-s + 10·25-s − 2·37-s + 11·49-s + 28·61-s + 20·73-s + 38·97-s + 38·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  − 2.77·13-s + 2·25-s − 0.328·37-s + 11/7·49-s + 3.58·61-s + 2.34·73-s + 3.85·97-s + 3.63·109-s − 2·121-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 + 3.76·169-s + 0.0760·173-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 + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15116544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15116544\)    =    \(2^{8} \cdot 3^{10}\)
Sign: $1$
Analytic conductor: \(963.843\)
Root analytic conductor: \(5.57187\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15116544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.361179749\)
\(L(\frac12)\) \(\approx\) \(2.361179749\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.a_al
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.13.k_bz
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.a_ach
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.a_aes
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.79.a_afb
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.89.a_agw
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2} \) 2.97.abm_vj
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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

−8.665539042875274873056750348447, −8.416963170923571541082869928268, −7.75518421533519040946674359356, −7.52575193544606295970146718479, −7.13668568638155373234807499260, −7.04533586242340999746666343824, −6.42733369651222158320728151636, −6.25723759574105145418968551534, −5.48699053123705789383172323760, −5.14471324328776300412958484157, −4.95142969122764262924888752986, −4.74115199057845646833395970396, −3.98941359928894195860114952354, −3.75224571957846151204971727794, −3.04854431429324880386763263848, −2.71409612348653969035337592244, −2.12866043508923671343335719466, −2.08926203324091003929469616784, −0.873410222619377677133362177247, −0.56888979240443523485451004707, 0.56888979240443523485451004707, 0.873410222619377677133362177247, 2.08926203324091003929469616784, 2.12866043508923671343335719466, 2.71409612348653969035337592244, 3.04854431429324880386763263848, 3.75224571957846151204971727794, 3.98941359928894195860114952354, 4.74115199057845646833395970396, 4.95142969122764262924888752986, 5.14471324328776300412958484157, 5.48699053123705789383172323760, 6.25723759574105145418968551534, 6.42733369651222158320728151636, 7.04533586242340999746666343824, 7.13668568638155373234807499260, 7.52575193544606295970146718479, 7.75518421533519040946674359356, 8.416963170923571541082869928268, 8.665539042875274873056750348447

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