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Properties

Label 4-2816e2-1.1-c1e2-0-6
Degree $4$
Conductor $7929856$
Sign $1$
Analytic cond. $505.614$
Root an. cond. $4.74192$
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  − 8·7-s + 5·9-s − 16·17-s + 10·23-s + 25-s − 2·31-s + 12·41-s + 24·47-s + 34·49-s − 40·63-s − 10·71-s + 20·73-s + 4·79-s + 16·81-s + 10·89-s + 26·97-s + 32·103-s + 2·113-s + 128·119-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 80·153-s + ⋯
L(s)  = 1  − 3.02·7-s + 5/3·9-s − 3.88·17-s + 2.08·23-s + 1/5·25-s − 0.359·31-s + 1.87·41-s + 3.50·47-s + 34/7·49-s − 5.03·63-s − 1.18·71-s + 2.34·73-s + 0.450·79-s + 16/9·81-s + 1.05·89-s + 2.63·97-s + 3.15·103-s + 0.188·113-s + 11.7·119-s − 0.0909·121-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 − 6.46·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7929856\)    =    \(2^{16} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(505.614\)
Root analytic conductor: \(4.74192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7929856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.523183450\)
\(L(\frac12)\) \(\approx\) \(1.523183450\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.17.q_du
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.23.ak_ct
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.31.c_cl
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.37.a_acn
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \) 2.59.a_aef
61$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.61.a_aes
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.67.a_an
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.71.k_gl
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.79.ae_gg
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.83.a_agg
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.89.ak_hv
97$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.97.aba_nz
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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

−9.113943466552335041513281212769, −9.020224640978495106574542353385, −8.420175436172294189546635984748, −7.50720547562015193116669191754, −7.31084526764459981170296496896, −7.01013391649383306285126929956, −6.70416908514197399734706205147, −6.51769316207722534556774044022, −6.00185675616932434152050289569, −5.84553825948189248544894515911, −4.78331199158964104828168893468, −4.72123075062595145467840419124, −4.22609227841717990764451898010, −3.73778788281334142816937276793, −3.50284619749260147525015325103, −2.80782627188277772139681244553, −2.28818628310804036610994582720, −2.20835130084077110730260248257, −0.887982755739150167773539801239, −0.50953367031073064800275828243, 0.50953367031073064800275828243, 0.887982755739150167773539801239, 2.20835130084077110730260248257, 2.28818628310804036610994582720, 2.80782627188277772139681244553, 3.50284619749260147525015325103, 3.73778788281334142816937276793, 4.22609227841717990764451898010, 4.72123075062595145467840419124, 4.78331199158964104828168893468, 5.84553825948189248544894515911, 6.00185675616932434152050289569, 6.51769316207722534556774044022, 6.70416908514197399734706205147, 7.01013391649383306285126929956, 7.31084526764459981170296496896, 7.50720547562015193116669191754, 8.420175436172294189546635984748, 9.020224640978495106574542353385, 9.113943466552335041513281212769

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