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Properties

Label 4-3024e2-1.1-c1e2-0-3
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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 + 4·19-s + 7·25-s − 12·29-s − 4·31-s + 2·37-s − 6·47-s + 9·49-s − 24·53-s + 6·59-s − 18·83-s − 16·103-s − 14·109-s − 12·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.917·19-s + 7/5·25-s − 2.22·29-s − 0.718·31-s + 0.328·37-s − 0.875·47-s + 9/7·49-s − 3.29·53-s + 0.781·59-s − 1.97·83-s − 1.57·103-s − 1.34·109-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + 1.07·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4497143992\)
\(L(\frac12)\) \(\approx\) \(0.4497143992\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.5.a_ah
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.41.a_acd
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.43.a_adf
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.59.ag_ex
61$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.61.a_ao
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.71.a_by
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2^2$ \( 1 - 155 T^{2} + p^{2} T^{4} \) 2.79.a_afz
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.83.s_jn
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2^2$ \( 1 - 182 T^{2} + p^{2} T^{4} \) 2.97.a_aha
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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

−9.138248026449053302342377060069, −8.538225596250738746071656699791, −8.198738774309401000854670376699, −7.65317891872947567198260653330, −7.41603312725463411136300208463, −7.02887744805862734279190386725, −6.65617777905017750899702143223, −6.23953783063163088546270124137, −6.02108975144052243175831722757, −5.32943563341249127788960798635, −5.25598620863187273958643884337, −4.69808879210362399669480077006, −4.08498856869721507486668329926, −3.59034317131380380292723925032, −3.45789320107128695360130772864, −2.72681400540064032077794680292, −2.67087672183296198655144691848, −1.63474306267788138417370552630, −1.30537098157031757089803790129, −0.21353598480165498711523389896, 0.21353598480165498711523389896, 1.30537098157031757089803790129, 1.63474306267788138417370552630, 2.67087672183296198655144691848, 2.72681400540064032077794680292, 3.45789320107128695360130772864, 3.59034317131380380292723925032, 4.08498856869721507486668329926, 4.69808879210362399669480077006, 5.25598620863187273958643884337, 5.32943563341249127788960798635, 6.02108975144052243175831722757, 6.23953783063163088546270124137, 6.65617777905017750899702143223, 7.02887744805862734279190386725, 7.41603312725463411136300208463, 7.65317891872947567198260653330, 8.198738774309401000854670376699, 8.538225596250738746071656699791, 9.138248026449053302342377060069

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