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Properties

Label 4-756e2-1.1-c1e2-0-7
Degree $4$
Conductor $571536$
Sign $1$
Analytic cond. $36.4416$
Root an. cond. $2.45696$
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  − 4·7-s + 10·13-s − 8·19-s + 5·25-s + 7·31-s − 11·37-s + 10·43-s + 9·49-s + 13·61-s − 5·67-s + 10·73-s − 17·79-s − 40·91-s + 10·97-s + 13·103-s − 17·109-s + 11·121-s + 127-s + 131-s + 32·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 1.51·7-s + 2.77·13-s − 1.83·19-s + 25-s + 1.25·31-s − 1.80·37-s + 1.52·43-s + 9/7·49-s + 1.66·61-s − 0.610·67-s + 1.17·73-s − 1.91·79-s − 4.19·91-s + 1.01·97-s + 1.28·103-s − 1.62·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(571536\)    =    \(2^{4} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(36.4416\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 571536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.611875157\)
\(L(\frac12)\) \(\approx\) \(1.611875157\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.13.ak_bz
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.i_bt
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.53.a_acb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ak_bb
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.r_ic
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−10.47051467324555825501408218195, −10.36085352411742237833569783296, −9.706371124018261947577911616879, −9.183872721975917997357219882368, −8.666957185865336672839115854772, −8.639266726384180833585740260978, −8.242672039494663270255201750844, −7.47428265685463010285386319793, −6.81585722559007291562249764007, −6.61078330450288838466668901466, −6.17631029587674796795130793101, −5.93539379745497448383189653466, −5.29954957061760091761372181978, −4.52859105018188863138849443064, −3.91089356611170728681137737208, −3.71449201499983786824730959351, −3.06720014594391318978330541493, −2.48793272427293153632119263446, −1.55263706632232339358065974440, −0.68110339004898393384495676703, 0.68110339004898393384495676703, 1.55263706632232339358065974440, 2.48793272427293153632119263446, 3.06720014594391318978330541493, 3.71449201499983786824730959351, 3.91089356611170728681137737208, 4.52859105018188863138849443064, 5.29954957061760091761372181978, 5.93539379745497448383189653466, 6.17631029587674796795130793101, 6.61078330450288838466668901466, 6.81585722559007291562249764007, 7.47428265685463010285386319793, 8.242672039494663270255201750844, 8.639266726384180833585740260978, 8.666957185865336672839115854772, 9.183872721975917997357219882368, 9.706371124018261947577911616879, 10.36085352411742237833569783296, 10.47051467324555825501408218195

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