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Properties

Label 8-1166e4-1.1-c1e4-0-2
Degree $8$
Conductor $1.848\times 10^{12}$
Sign $1$
Analytic cond. $7514.54$
Root an. cond. $3.05132$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s − 4·5-s − 16·6-s − 4·7-s + 20·8-s + 4·9-s − 16·10-s − 4·11-s − 40·12-s − 8·13-s − 16·14-s + 16·15-s + 35·16-s + 16·18-s − 8·19-s − 40·20-s + 16·21-s − 16·22-s − 8·23-s − 80·24-s − 32·26-s + 8·27-s − 40·28-s − 4·29-s + 64·30-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s − 1.51·7-s + 7.07·8-s + 4/3·9-s − 5.05·10-s − 1.20·11-s − 11.5·12-s − 2.21·13-s − 4.27·14-s + 4.13·15-s + 35/4·16-s + 3.77·18-s − 1.83·19-s − 8.94·20-s + 3.49·21-s − 3.41·22-s − 1.66·23-s − 16.3·24-s − 6.27·26-s + 1.53·27-s − 7.55·28-s − 0.742·29-s + 11.6·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 53^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{4} \cdot 53^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 11^{4} \cdot 53^{4}\)
Sign: $1$
Analytic conductor: \(7514.54\)
Root analytic conductor: \(3.05132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 11^{4} \cdot 53^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
53$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 4 p T^{2} + 8 p T^{3} + 47 T^{4} + 8 p^{2} T^{5} + 4 p^{3} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.3.e_m_y_bv
5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.5.e_q_bs_eo
7$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 18 T^{2} + 32 T^{3} + 108 T^{4} + 32 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.7.e_s_bg_ee
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 50 T^{2} + 16 p T^{3} + 811 T^{4} + 16 p^{2} T^{5} + 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.13.i_by_ia_bff
17$C_2 \wr C_2\wr C_2$ \( 1 + 60 T^{2} + 8 T^{3} + 1463 T^{4} + 8 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \) 4.17.a_ci_i_ceh
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 82 T^{2} + 384 T^{3} + 2267 T^{4} + 384 p T^{5} + 82 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.19.i_de_ou_djf
23$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 86 T^{2} + 400 T^{3} + 2635 T^{4} + 400 p T^{5} + 86 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.23.i_di_pk_dxj
29$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 86 T^{2} + 344 T^{3} + 3307 T^{4} + 344 p T^{5} + 86 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.29.e_di_ng_exf
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 116 T^{2} + 28 p T^{3} + 5562 T^{4} + 28 p^{2} T^{5} + 116 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.31.m_em_bhk_ify
37$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 140 T^{2} + 808 T^{3} + 6063 T^{4} + 808 p T^{5} + 140 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.37.m_fk_bfc_izf
41$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 138 T^{2} - 392 T^{3} + 7956 T^{4} - 392 p T^{5} + 138 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.41.ae_fi_apc_lua
43$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 162 T^{2} + 1200 T^{3} + 9836 T^{4} + 1200 p T^{5} + 162 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.43.m_gg_bue_ooi
47$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 190 T^{2} + 1508 T^{3} + 11028 T^{4} + 1508 p T^{5} + 190 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.47.q_hi_cga_qie
59$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 358 T^{2} + 3732 T^{3} + 31524 T^{4} + 3732 p T^{5} + 358 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) 4.59.y_nu_fno_buqm
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 212 T^{2} + 1648 T^{3} + 14838 T^{4} + 1648 p T^{5} + 212 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.61.q_ie_clk_vys
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 92 T^{2} - 72 T^{3} + 2454 T^{4} - 72 p T^{5} + 92 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.67.ai_do_acu_dqk
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 198 T^{2} + 1808 T^{3} + 18203 T^{4} + 1808 p T^{5} + 198 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.71.i_hq_cro_bayd
73$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 240 T^{2} - 1812 T^{3} + 23366 T^{4} - 1812 p T^{5} + 240 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) 4.73.am_jg_acrs_bios
79$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 428 T^{2} + 5280 T^{3} + 54391 T^{4} + 5280 p T^{5} + 428 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) 4.79.y_qm_hvc_dclz
83$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 262 T^{2} - 656 T^{3} + 29323 T^{4} - 656 p T^{5} + 262 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) 4.83.ae_kc_azg_brjv
89$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 252 T^{2} + 2236 T^{3} + 31130 T^{4} + 2236 p T^{5} + 252 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.89.m_js_dia_bubi
97$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 590 T^{2} - 7696 T^{3} + 96531 T^{4} - 7696 p T^{5} + 590 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) 4.97.ay_ws_alka_fmut
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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

−7.23828387069139891836400247672, −7.00085965890925907876184251213, −6.97930637191525399704719343903, −6.48527332933510221182420959823, −6.47801638735345248959894680745, −6.08603429940528089581779912727, −6.05050228538609457006329249064, −5.78909773706051967591783275101, −5.59505098001844834089791511133, −5.57215602586069045388628057643, −5.06612746972371222055114435092, −4.96771877854418079644519412917, −4.74706283499780208524599744802, −4.64785373464415625913076628407, −4.27307208406513996242468245535, −4.16670121171276111545589134805, −3.85428941144818672311456361618, −3.33186343393825134493048519663, −3.29709533371791347316714509514, −3.24504681387276748045761246295, −3.00636714041444864063573549346, −2.36394023203127983512396031949, −2.16328800715794136509933062114, −1.84285753288551225307655186356, −1.65336785782173664744825111586, 0, 0, 0, 0, 1.65336785782173664744825111586, 1.84285753288551225307655186356, 2.16328800715794136509933062114, 2.36394023203127983512396031949, 3.00636714041444864063573549346, 3.24504681387276748045761246295, 3.29709533371791347316714509514, 3.33186343393825134493048519663, 3.85428941144818672311456361618, 4.16670121171276111545589134805, 4.27307208406513996242468245535, 4.64785373464415625913076628407, 4.74706283499780208524599744802, 4.96771877854418079644519412917, 5.06612746972371222055114435092, 5.57215602586069045388628057643, 5.59505098001844834089791511133, 5.78909773706051967591783275101, 6.05050228538609457006329249064, 6.08603429940528089581779912727, 6.47801638735345248959894680745, 6.48527332933510221182420959823, 6.97930637191525399704719343903, 7.00085965890925907876184251213, 7.23828387069139891836400247672

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