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Properties

Label 8-3328e4-1.1-c1e4-0-11
Degree $8$
Conductor $1.227\times 10^{14}$
Sign $1$
Analytic cond. $498702.$
Root an. cond. $5.15501$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 2·5-s − 9-s + 2·11-s − 4·13-s − 4·15-s − 2·17-s + 6·19-s − 12·23-s − 9·25-s + 2·27-s − 16·29-s − 10·31-s − 4·33-s + 14·37-s + 8·39-s − 20·41-s + 14·43-s − 2·45-s − 20·47-s − 15·49-s + 4·51-s − 4·53-s + 4·55-s − 12·57-s + 2·59-s − 8·65-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1/3·9-s + 0.603·11-s − 1.10·13-s − 1.03·15-s − 0.485·17-s + 1.37·19-s − 2.50·23-s − 9/5·25-s + 0.384·27-s − 2.97·29-s − 1.79·31-s − 0.696·33-s + 2.30·37-s + 1.28·39-s − 3.12·41-s + 2.13·43-s − 0.298·45-s − 2.91·47-s − 2.14·49-s + 0.560·51-s − 0.549·53-s + 0.539·55-s − 1.58·57-s + 0.260·59-s − 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(498702.\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{32} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
13$C_1$ \( ( 1 + T )^{4} \)
good3$D_4\times C_2$ \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) 4.3.c_f_k_q
5$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 13 T^{2} - 22 T^{3} + 84 T^{4} - 22 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.5.ac_n_aw_dg
7$C_2 \wr C_2\wr C_2$ \( 1 + 15 T^{2} - 18 T^{3} + 110 T^{4} - 18 p T^{5} + 15 p^{2} T^{6} + p^{4} T^{8} \) 4.7.a_p_as_eg
11$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 34 T^{2} - 58 T^{3} + 522 T^{4} - 58 p T^{5} + 34 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.11.ac_bi_acg_uc
17$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 29 T^{2} - 46 T^{3} + 304 T^{4} - 46 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) 4.17.c_bd_abu_ls
19$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 42 T^{2} - 198 T^{3} + 938 T^{4} - 198 p T^{5} + 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) 4.19.ag_bq_ahq_bkc
23$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 100 T^{2} + 636 T^{3} + 3414 T^{4} + 636 p T^{5} + 100 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.23.m_dw_ym_fbi
29$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 160 T^{2} + 1088 T^{3} + 6414 T^{4} + 1088 p T^{5} + 160 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.29.q_ge_bpw_jms
31$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 90 T^{2} + 338 T^{3} + 2186 T^{4} + 338 p T^{5} + 90 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) 4.31.k_dm_na_dgc
37$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 181 T^{2} - 1334 T^{3} + 9892 T^{4} - 1334 p T^{5} + 181 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) 4.37.ao_gz_abzi_oqm
41$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 268 T^{2} + 2476 T^{3} + 18246 T^{4} + 2476 p T^{5} + 268 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) 4.41.u_ki_drg_bazu
43$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 165 T^{2} - 1546 T^{3} + 10928 T^{4} - 1546 p T^{5} + 165 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) 4.43.ao_gj_achm_qei
47$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 311 T^{2} + 3014 T^{3} + 24622 T^{4} + 3014 p T^{5} + 311 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) 4.47.u_lz_ely_bkla
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 124 T^{2} + 380 T^{3} + 7398 T^{4} + 380 p T^{5} + 124 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.53.e_eu_oq_kyo
59$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 206 T^{2} - 314 T^{3} + 17530 T^{4} - 314 p T^{5} + 206 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) 4.59.ac_hy_amc_zyg
61$C_2 \wr C_2\wr C_2$ \( 1 + 136 T^{2} - 288 T^{3} + 9726 T^{4} - 288 p T^{5} + 136 p^{2} T^{6} + p^{4} T^{8} \) 4.61.a_fg_alc_okc
67$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 198 T^{2} + 1110 T^{3} + 17546 T^{4} + 1110 p T^{5} + 198 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) 4.67.g_hq_bqs_zyw
71$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 551 T^{2} + 6994 T^{3} + 69526 T^{4} + 6994 p T^{5} + 551 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) 4.71.bc_vf_kja_dywc
73$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 156 T^{2} - 1144 T^{3} + 17702 T^{4} - 1144 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.73.ai_ga_absa_baew
79$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 312 T^{2} + 3680 T^{3} + 36782 T^{4} + 3680 p T^{5} + 312 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.79.q_ma_flo_ccks
83$C_2 \wr C_2\wr C_2$ \( 1 - 22 T + 430 T^{2} - 5366 T^{3} + 56730 T^{4} - 5366 p T^{5} + 430 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) 4.83.aw_qo_ahyk_dfxy
89$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 328 T^{2} - 2104 T^{3} + 42606 T^{4} - 2104 p T^{5} + 328 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.89.ai_mq_adcy_clas
97$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 480 T^{2} + 5932 T^{3} + 74270 T^{4} + 5932 p T^{5} + 480 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) 4.97.u_sm_iue_efwo
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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

−6.43182931962522817846395227564, −6.12539707830265304708632919229, −5.94941705042509004252694096419, −5.93205182042912648449036486950, −5.84221206021227140821518836849, −5.51191118484757045023548582316, −5.34654314261061689774237988445, −5.21806467297970713608256452879, −5.07843993292595166221174889747, −4.74537287375082116052868145264, −4.56566673689254170683454961049, −4.19194645342560880562075080533, −4.10544405636262741288652707730, −3.79278645322855120072718611002, −3.71370869506734098716273205003, −3.37833837145546457452286958330, −3.31983915704010559924477838847, −2.80025026364366820690201664984, −2.70215688562403743065719930856, −2.20326758239875656760264500051, −2.20305618287858281218525621050, −1.79865744636110240077933668531, −1.68401005099443050618748064353, −1.33114097664460113138778486405, −1.28422894586369504566290146681, 0, 0, 0, 0, 1.28422894586369504566290146681, 1.33114097664460113138778486405, 1.68401005099443050618748064353, 1.79865744636110240077933668531, 2.20305618287858281218525621050, 2.20326758239875656760264500051, 2.70215688562403743065719930856, 2.80025026364366820690201664984, 3.31983915704010559924477838847, 3.37833837145546457452286958330, 3.71370869506734098716273205003, 3.79278645322855120072718611002, 4.10544405636262741288652707730, 4.19194645342560880562075080533, 4.56566673689254170683454961049, 4.74537287375082116052868145264, 5.07843993292595166221174889747, 5.21806467297970713608256452879, 5.34654314261061689774237988445, 5.51191118484757045023548582316, 5.84221206021227140821518836849, 5.93205182042912648449036486950, 5.94941705042509004252694096419, 6.12539707830265304708632919229, 6.43182931962522817846395227564

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