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Properties

Label 4-3024e2-1.1-c1e2-0-44
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·7-s + 4·11-s + 2·19-s + 8·23-s − 3·25-s − 12·29-s + 6·31-s + 6·37-s − 12·41-s + 4·43-s + 12·47-s + 3·49-s + 4·59-s + 8·61-s − 8·67-s + 12·73-s + 8·77-s + 12·79-s + 20·83-s + 12·89-s + 16·97-s − 8·101-s − 2·103-s + 16·107-s − 2·109-s − 8·113-s − 3·121-s + ⋯
L(s)  = 1  + 0.755·7-s + 1.20·11-s + 0.458·19-s + 1.66·23-s − 3/5·25-s − 2.22·29-s + 1.07·31-s + 0.986·37-s − 1.87·41-s + 0.609·43-s + 1.75·47-s + 3/7·49-s + 0.520·59-s + 1.02·61-s − 0.977·67-s + 1.40·73-s + 0.911·77-s + 1.35·79-s + 2.19·83-s + 1.27·89-s + 1.62·97-s − 0.796·101-s − 0.197·103-s + 1.54·107-s − 0.191·109-s − 0.752·113-s − 0.272·121-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

\(L(1)\) \(\approx\) \(3.883159966\)
\(L(\frac12)\) \(\approx\) \(3.883159966\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
11$D_{4}$ \( 1 - 4 T + 19 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_t
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$D_{4}$ \( 1 - 8 T + 55 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_cd
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.31.ag_br
37$D_{4}$ \( 1 - 6 T + 55 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_cd
41$D_{4}$ \( 1 + 12 T + 111 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_eh
43$D_{4}$ \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_ck
47$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_dy
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$D_{4}$ \( 1 - 4 T + 94 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.59.ae_dq
61$D_{4}$ \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_ba
67$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_es
71$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.71.a_abh
73$D_{4}$ \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_fy
79$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.79.am_gk
83$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.83.au_kg
89$D_{4}$ \( 1 - 12 T + 151 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_fv
97$D_{4}$ \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.97.aq_fq
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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

−8.930652036205161120265365926432, −8.729119403403450663487493329435, −7.975725736994327291797997289621, −7.83672587032887635920019989122, −7.49109259480459555614805427469, −7.04758861008663795749748082928, −6.53998501095229081061482730522, −6.49175830333650530008332078977, −5.64992863100969220083769518933, −5.61545019504507832384418811726, −4.96019921038343193702910704191, −4.78179414320327753104482914099, −4.11487970698589143607449270346, −3.81421960798734821151372032567, −3.42148458654315459055090734582, −2.89173825206231459120671006537, −2.04523566260008101835170603536, −2.00287651010499383647707168627, −1.05974916155509297642056259261, −0.74807549459243729831945013422, 0.74807549459243729831945013422, 1.05974916155509297642056259261, 2.00287651010499383647707168627, 2.04523566260008101835170603536, 2.89173825206231459120671006537, 3.42148458654315459055090734582, 3.81421960798734821151372032567, 4.11487970698589143607449270346, 4.78179414320327753104482914099, 4.96019921038343193702910704191, 5.61545019504507832384418811726, 5.64992863100969220083769518933, 6.49175830333650530008332078977, 6.53998501095229081061482730522, 7.04758861008663795749748082928, 7.49109259480459555614805427469, 7.83672587032887635920019989122, 7.975725736994327291797997289621, 8.729119403403450663487493329435, 8.930652036205161120265365926432

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