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Properties

Label 8-1656e4-1.1-c1e4-0-1
Degree $8$
Conductor $7.520\times 10^{12}$
Sign $1$
Analytic cond. $30573.8$
Root an. cond. $3.63637$
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  + 4·5-s + 2·7-s + 2·11-s + 4·13-s + 2·17-s + 8·19-s − 4·23-s + 4·25-s + 12·29-s + 12·31-s + 8·35-s + 14·37-s + 4·41-s + 4·43-s − 12·47-s + 2·49-s + 8·53-s + 8·55-s − 16·59-s + 6·61-s + 16·65-s + 8·67-s − 12·71-s + 16·73-s + 4·77-s + 10·79-s − 14·83-s + ⋯
L(s)  = 1  + 1.78·5-s + 0.755·7-s + 0.603·11-s + 1.10·13-s + 0.485·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s + 2.22·29-s + 2.15·31-s + 1.35·35-s + 2.30·37-s + 0.624·41-s + 0.609·43-s − 1.75·47-s + 2/7·49-s + 1.09·53-s + 1.07·55-s − 2.08·59-s + 0.768·61-s + 1.98·65-s + 0.977·67-s − 1.42·71-s + 1.87·73-s + 0.455·77-s + 1.12·79-s − 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(30573.8\)
Root analytic conductor: \(3.63637\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(13.67456375\)
\(L(\frac12)\) \(\approx\) \(13.67456375\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_1$ \( ( 1 + T )^{4} \)
good5$D_{4}$ \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) 4.5.ae_m_abk_ec
7$(C_4^2 : C_2):C_2$ \( 1 - 2 T + 2 T^{2} + 6 T^{3} + 2 T^{4} + 6 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.7.ac_c_g_c
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 2 p T^{2} + 6 T^{3} + 194 T^{4} + 6 p T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.11.ac_w_g_hm
13$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 32 T^{2} - 108 T^{3} + 526 T^{4} - 108 p T^{5} + 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.13.ae_bg_aee_ug
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 46 T^{2} - 30 T^{3} + 938 T^{4} - 30 p T^{5} + 46 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.17.ac_bu_abe_bkc
19$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 60 T^{2} - 216 T^{3} + 1114 T^{4} - 216 p T^{5} + 60 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.19.ai_ci_aii_bqw
29$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 144 T^{2} - 996 T^{3} + 6574 T^{4} - 996 p T^{5} + 144 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) 4.29.am_fo_abmi_jsw
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 152 T^{2} - 1068 T^{3} + 7406 T^{4} - 1068 p T^{5} + 152 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) 4.31.am_fw_abpc_kyw
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 138 T^{2} - 938 T^{3} + 5882 T^{4} - 938 p T^{5} + 138 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) 4.37.ao_fi_abkc_isg
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 88 T^{2} - 108 T^{3} + 3662 T^{4} - 108 p T^{5} + 88 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.41.ae_dk_aee_fkw
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 28 T^{2} - 220 T^{3} + 3178 T^{4} - 220 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.43.ae_bc_aim_esg
47$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 132 T^{2} + 972 T^{3} + 8006 T^{4} + 972 p T^{5} + 132 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.47.m_fc_blk_lvy
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 140 T^{2} - 920 T^{3} + 10906 T^{4} - 920 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.53.ai_fk_abjk_qdm
59$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 144 T^{2} + 912 T^{3} + 5166 T^{4} + 912 p T^{5} + 144 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.59.q_fo_bjc_hqs
61$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 234 T^{2} - 1090 T^{3} + 21114 T^{4} - 1090 p T^{5} + 234 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) 4.61.ag_ja_abpy_bfgc
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 140 T^{2} - 696 T^{3} + 10906 T^{4} - 696 p T^{5} + 140 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.67.ai_fk_abau_qdm
71$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 256 T^{2} + 2396 T^{3} + 26398 T^{4} + 2396 p T^{5} + 256 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.71.m_jw_doe_bnbi
73$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 224 T^{2} - 2112 T^{3} + 21950 T^{4} - 2112 p T^{5} + 224 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) 4.73.aq_iq_addg_bgmg
79$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 282 T^{2} - 2082 T^{3} + 32722 T^{4} - 2082 p T^{5} + 282 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) 4.79.ak_kw_adcc_bwko
83$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 382 T^{2} + 3542 T^{3} + 49650 T^{4} + 3542 p T^{5} + 382 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) 4.83.o_os_fgg_cvlq
89$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 230 T^{2} - 2954 T^{3} + 28234 T^{4} - 2954 p T^{5} + 230 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) 4.89.ao_iw_aejq_bpty
97$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 284 T^{2} - 1116 T^{3} + 37766 T^{4} - 1116 p T^{5} + 284 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.97.ae_ky_abqy_cdwo
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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

−6.59320612457478873234198813473, −6.31268843364268442784566980312, −6.20564387004907465705794255910, −6.05959431853814355327209549084, −5.99822305734818940966006542571, −5.49816395681240387614524637422, −5.42261032142927870528859413032, −5.37217948918273670226244695010, −4.96093533312459127612412212047, −4.50479442032574763821260944168, −4.50166017300869695527825766997, −4.41656929300462880184131312178, −4.27087981039498300008907132362, −3.55726625378321568148527032775, −3.38166622553015718928434542083, −3.34668603118406206643199305416, −3.06376921283709942407792889113, −2.51433896368408822046420098852, −2.48929369318043303464400810667, −2.15351378088906801295914843386, −1.91184610810155729184832578275, −1.45983876810372015584272855720, −1.05207826399310462125380514504, −1.00921544431099807793671284003, −0.76321128442087281344768248191, 0.76321128442087281344768248191, 1.00921544431099807793671284003, 1.05207826399310462125380514504, 1.45983876810372015584272855720, 1.91184610810155729184832578275, 2.15351378088906801295914843386, 2.48929369318043303464400810667, 2.51433896368408822046420098852, 3.06376921283709942407792889113, 3.34668603118406206643199305416, 3.38166622553015718928434542083, 3.55726625378321568148527032775, 4.27087981039498300008907132362, 4.41656929300462880184131312178, 4.50166017300869695527825766997, 4.50479442032574763821260944168, 4.96093533312459127612412212047, 5.37217948918273670226244695010, 5.42261032142927870528859413032, 5.49816395681240387614524637422, 5.99822305734818940966006542571, 6.05959431853814355327209549084, 6.20564387004907465705794255910, 6.31268843364268442784566980312, 6.59320612457478873234198813473

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