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Properties

Label 4-4032e2-1.1-c1e2-0-5
Degree $4$
Conductor $16257024$
Sign $1$
Analytic cond. $1036.56$
Root an. cond. $5.67412$
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  + 10·25-s − 4·29-s − 12·37-s − 7·49-s − 20·53-s + 36·109-s − 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯
L(s)  = 1  + 2·25-s − 0.742·29-s − 1.97·37-s − 49-s − 2.74·53-s + 3.44·109-s − 0.376·113-s − 0.545·121-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 + 0.0747·179-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 + 0.0669·223-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(16257024\)    =    \(2^{12} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1036.56\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 16257024,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.648168200\)
\(L(\frac12)\) \(\approx\) \(1.648168200\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.a_aek
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
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$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−8.609574947517546075962265079641, −8.411715803373460716983072648287, −7.70904374231593601463597270838, −7.68810055061322622935243639120, −7.13279832378630315776845523664, −6.72757210428033329034350247874, −6.52701038884928792590406135849, −6.14612933323413287775395457128, −5.54077655308713678843217833674, −5.30617935676239931266497629420, −4.80396695372781443002574279281, −4.61390478063896448842881982765, −4.10490703329801993545648018676, −3.36588401064525294584337415258, −3.29792091092080667278049004127, −2.88198313905676551098254086385, −2.10629265148899205655608464761, −1.74118179592128553306189160726, −1.19128981666386702921935889814, −0.38360277681249166375749790802, 0.38360277681249166375749790802, 1.19128981666386702921935889814, 1.74118179592128553306189160726, 2.10629265148899205655608464761, 2.88198313905676551098254086385, 3.29792091092080667278049004127, 3.36588401064525294584337415258, 4.10490703329801993545648018676, 4.61390478063896448842881982765, 4.80396695372781443002574279281, 5.30617935676239931266497629420, 5.54077655308713678843217833674, 6.14612933323413287775395457128, 6.52701038884928792590406135849, 6.72757210428033329034350247874, 7.13279832378630315776845523664, 7.68810055061322622935243639120, 7.70904374231593601463597270838, 8.411715803373460716983072648287, 8.609574947517546075962265079641

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