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Properties

Label 4-3024e2-1.1-c1e2-0-43
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  + 2·5-s − 7-s + 10·11-s + 5·13-s + 3·17-s + 19-s + 6·23-s − 7·25-s − 29-s − 2·35-s − 3·37-s − 5·41-s − 43-s − 6·49-s − 9·53-s + 20·55-s + 14·61-s + 10·65-s + 4·67-s − 24·71-s − 3·73-s − 10·77-s + 8·79-s + 9·83-s + 6·85-s − 13·89-s − 5·91-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s + 3.01·11-s + 1.38·13-s + 0.727·17-s + 0.229·19-s + 1.25·23-s − 7/5·25-s − 0.185·29-s − 0.338·35-s − 0.493·37-s − 0.780·41-s − 0.152·43-s − 6/7·49-s − 1.23·53-s + 2.69·55-s + 1.79·61-s + 1.24·65-s + 0.488·67-s − 2.84·71-s − 0.351·73-s − 1.13·77-s + 0.900·79-s + 0.987·83-s + 0.650·85-s − 1.37·89-s − 0.524·91-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

\(L(1)\) \(\approx\) \(4.737056573\)
\(L(\frac12)\) \(\approx\) \(4.737056573\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.af_m
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.17.ad_ai
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2^2$ \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_abc
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_aq
43$C_2^2$ \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) 2.43.b_abq
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.53.j_bc
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ao_ff
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_abz
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 + 3 T - 64 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.73.d_acm
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.79.ai_ap
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.83.aj_ac
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.89.n_dc
97$C_2^2$ \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_aq
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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

−8.918859199082795871664423579115, −8.590365134835161143553989353765, −8.425163263598372378745090757181, −7.76057726296778194354212145552, −7.20565101034538640035239167109, −7.05660037043037595485967649950, −6.43329638643278680683277426974, −6.29469650194635748009625564840, −6.03455711531137230132642479084, −5.64511206914246562437083731446, −5.10379997024218611505379625004, −4.61654783257123460182079070834, −4.07095224527922946114653834858, −3.71871859503084967345472126804, −3.36813166160613424294643481135, −3.10973874956882399641923347473, −2.09543035483855014377640214733, −1.62170973194198831953417429904, −1.37727501355599320080280072529, −0.74279662652149815704298699566, 0.74279662652149815704298699566, 1.37727501355599320080280072529, 1.62170973194198831953417429904, 2.09543035483855014377640214733, 3.10973874956882399641923347473, 3.36813166160613424294643481135, 3.71871859503084967345472126804, 4.07095224527922946114653834858, 4.61654783257123460182079070834, 5.10379997024218611505379625004, 5.64511206914246562437083731446, 6.03455711531137230132642479084, 6.29469650194635748009625564840, 6.43329638643278680683277426974, 7.05660037043037595485967649950, 7.20565101034538640035239167109, 7.76057726296778194354212145552, 8.425163263598372378745090757181, 8.590365134835161143553989353765, 8.918859199082795871664423579115

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