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Properties

Label 8-5445e4-1.1-c1e4-0-6
Degree $8$
Conductor $8.790\times 10^{14}$
Sign $1$
Analytic cond. $3.57355\times 10^{6}$
Root an. cond. $6.59382$
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  − 3·2-s + 4·4-s − 4·5-s + 11·7-s − 4·8-s + 12·10-s + 7·13-s − 33·14-s + 2·16-s − 3·17-s + 12·19-s − 16·20-s + 9·23-s + 10·25-s − 21·26-s + 44·28-s + 8·29-s + 3·31-s + 5·32-s + 9·34-s − 44·35-s − 3·37-s − 36·38-s + 16·40-s + 7·41-s + 21·43-s − 27·46-s + ⋯
L(s)  = 1  − 2.12·2-s + 2·4-s − 1.78·5-s + 4.15·7-s − 1.41·8-s + 3.79·10-s + 1.94·13-s − 8.81·14-s + 1/2·16-s − 0.727·17-s + 2.75·19-s − 3.57·20-s + 1.87·23-s + 2·25-s − 4.11·26-s + 8.31·28-s + 1.48·29-s + 0.538·31-s + 0.883·32-s + 1.54·34-s − 7.43·35-s − 0.493·37-s − 5.83·38-s + 2.52·40-s + 1.09·41-s + 3.20·43-s − 3.98·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.57355\times 10^{6}\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.652917491\)
\(L(\frac12)\) \(\approx\) \(5.652917491\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good2$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 11 T^{4} + 7 p T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) 4.2.d_f_h_l
7$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 67 T^{2} - 276 T^{3} + 845 T^{4} - 276 p T^{5} + 67 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) 4.7.al_cp_akq_bgn
13$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 59 T^{2} - 264 T^{3} + 1185 T^{4} - 264 p T^{5} + 59 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) 4.13.ah_ch_ake_btp
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 60 T^{2} + 127 T^{3} + 1451 T^{4} + 127 p T^{5} + 60 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) 4.17.d_ci_ex_cdv
19$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 122 T^{2} - 749 T^{3} + 3939 T^{4} - 749 p T^{5} + 122 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) 4.19.am_es_abcv_fvn
23$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 38 T^{2} + 85 T^{3} - 979 T^{4} + 85 p T^{5} + 38 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) 4.23.aj_bm_dh_ablr
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 112 T^{2} - 601 T^{3} + 4759 T^{4} - 601 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.29.ai_ei_axd_hbb
31$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 276 T^{3} + 3945 T^{4} - 276 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) 4.31.ad_dp_akq_fvt
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 92 T^{2} + 305 T^{3} + 4221 T^{4} + 305 p T^{5} + 92 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) 4.37.d_do_lt_ggj
41$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 118 T^{2} - 479 T^{3} + 5815 T^{4} - 479 p T^{5} + 118 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) 4.41.ah_eo_asl_ipr
43$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 293 T^{2} - 2900 T^{3} + 21441 T^{4} - 2900 p T^{5} + 293 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) 4.43.av_lh_aeho_bfsr
47$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 137 T^{2} - 290 T^{3} + 8531 T^{4} - 290 p T^{5} + 137 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) 4.47.ad_fh_ale_mqd
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 245 T^{2} - 1776 T^{3} + 20351 T^{4} - 1776 p T^{5} + 245 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) 4.53.al_jl_acqi_bect
59$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 127 T^{2} - 44 T^{3} + 4999 T^{4} - 44 p T^{5} + 127 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) 4.59.ah_ex_abs_hkh
61$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 203 T^{2} - 642 T^{3} + 17379 T^{4} - 642 p T^{5} + 203 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.61.ae_hv_ays_zsl
67$C_2 \wr C_2\wr C_2$ \( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) 4.67.b_he_abl_xll
71$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 186 T^{2} - 925 T^{3} + 8531 T^{4} - 925 p T^{5} + 186 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) 4.71.ap_he_abjp_mqd
73$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 218 T^{2} + 1375 T^{3} + 20781 T^{4} + 1375 p T^{5} + 218 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) 4.73.j_ik_cax_beth
79$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 220 T^{2} - 487 T^{3} + 20123 T^{4} - 487 p T^{5} + 220 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) 4.79.ag_im_ast_bdtz
83$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 375 T^{2} + 3710 T^{3} + 48443 T^{4} + 3710 p T^{5} + 375 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) 4.83.p_ol_fms_ctrf
89$C_2 \wr C_2\wr C_2$ \( 1 + 206 T^{2} + 400 T^{3} + 21551 T^{4} + 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} \) 4.89.a_hy_pk_bfwx
97$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 332 T^{2} - 1836 T^{3} + 45565 T^{4} - 1836 p T^{5} + 332 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) 4.97.ag_mu_acsq_cpkn
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.71672685156435953928867777817, −5.58903190232855288554176393831, −5.38728749221506580554892754537, −5.06116716188037074351364246437, −5.03614478268394401631668416953, −4.63571977979730669016563328123, −4.55174419030441546139555751056, −4.46804299721849448996765373690, −4.40880596237179024512541941274, −3.96923297466112712179292842343, −3.90006700988123738505218084562, −3.50086235211323205171046149617, −3.45455165585123937847621035627, −3.00619173003503707513392210975, −2.89177726265964733064392253151, −2.63560409527955333906321116294, −2.38710037690402568202614815763, −2.16078236399228118199532500964, −1.66580590126260219096892195017, −1.57946493967274663139089616525, −1.36754505697492513457452004947, −0.964628749644443913829384524833, −0.77253663345412366708555635398, −0.75472824504636374677857572188, −0.64851481299306459040536794654, 0.64851481299306459040536794654, 0.75472824504636374677857572188, 0.77253663345412366708555635398, 0.964628749644443913829384524833, 1.36754505697492513457452004947, 1.57946493967274663139089616525, 1.66580590126260219096892195017, 2.16078236399228118199532500964, 2.38710037690402568202614815763, 2.63560409527955333906321116294, 2.89177726265964733064392253151, 3.00619173003503707513392210975, 3.45455165585123937847621035627, 3.50086235211323205171046149617, 3.90006700988123738505218084562, 3.96923297466112712179292842343, 4.40880596237179024512541941274, 4.46804299721849448996765373690, 4.55174419030441546139555751056, 4.63571977979730669016563328123, 5.03614478268394401631668416953, 5.06116716188037074351364246437, 5.38728749221506580554892754537, 5.58903190232855288554176393831, 5.71672685156435953928867777817

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