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Properties

Label 4-2816e2-1.1-c1e2-0-5
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  − 3·9-s − 8·17-s + 6·23-s + 9·25-s + 18·31-s + 4·41-s − 24·47-s − 14·49-s − 6·71-s + 12·73-s + 12·79-s + 10·89-s − 6·97-s + 2·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯
L(s)  = 1  − 9-s − 1.94·17-s + 1.25·23-s + 9/5·25-s + 3.23·31-s + 0.624·41-s − 3.50·47-s − 2·49-s − 0.712·71-s + 1.40·73-s + 1.35·79-s + 1.05·89-s − 0.609·97-s + 0.188·113-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-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

\(L(1)\) \(\approx\) \(1.706864350\)
\(L(\frac12)\) \(\approx\) \(1.706864350\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.3.a_d
5$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.5.a_aj
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.31.as_fn
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
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.y_je
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 - 37 T^{2} + p^{2} T^{4} \) 2.59.a_abl
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \) 2.67.a_dn
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.71.g_fv
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.79.am_hm
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.89.ak_hv
97$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.97.g_hv
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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

−9.392086100499182561755264752734, −8.750105836294209663413163887679, −8.315054346767562433101291700044, −8.005789743318480357438719149980, −7.55659931333735151236065809675, −6.82977912880411798063043168642, −6.65307895519457746416621411038, −6.38118540008766124118658620522, −6.22259317571853782387833088253, −5.38967428685759058803036258059, −4.91260001559214617069605785547, −4.73170526063415830280671752970, −4.56250614823487720188118352618, −3.75627577642522841313900165017, −3.07798658763287426211704071760, −2.97491128041184255148985677765, −2.52145535295954422830229438274, −1.86134422251170874896607481034, −1.14513774228869709168742972294, −0.45663135296924132521760790629, 0.45663135296924132521760790629, 1.14513774228869709168742972294, 1.86134422251170874896607481034, 2.52145535295954422830229438274, 2.97491128041184255148985677765, 3.07798658763287426211704071760, 3.75627577642522841313900165017, 4.56250614823487720188118352618, 4.73170526063415830280671752970, 4.91260001559214617069605785547, 5.38967428685759058803036258059, 6.22259317571853782387833088253, 6.38118540008766124118658620522, 6.65307895519457746416621411038, 6.82977912880411798063043168642, 7.55659931333735151236065809675, 8.005789743318480357438719149980, 8.315054346767562433101291700044, 8.750105836294209663413163887679, 9.392086100499182561755264752734

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