Properties

Label 8-7406e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.008\times 10^{15}$
Sign $1$
Analytic cond. $1.22304\times 10^{7}$
Root an. cond. $7.69007$
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  − 4·2-s − 4·3-s + 10·4-s + 16·6-s − 4·7-s − 20·8-s + 2·9-s − 8·11-s − 40·12-s − 4·13-s + 16·14-s + 35·16-s + 8·17-s − 8·18-s − 8·19-s + 16·21-s + 32·22-s + 80·24-s − 10·25-s + 16·26-s + 16·27-s − 40·28-s − 4·29-s − 4·31-s − 56·32-s + 32·33-s − 32·34-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 1.51·7-s − 7.07·8-s + 2/3·9-s − 2.41·11-s − 11.5·12-s − 1.10·13-s + 4.27·14-s + 35/4·16-s + 1.94·17-s − 1.88·18-s − 1.83·19-s + 3.49·21-s + 6.82·22-s + 16.3·24-s − 2·25-s + 3.13·26-s + 3.07·27-s − 7.55·28-s − 0.742·29-s − 0.718·31-s − 9.89·32-s + 5.57·33-s − 5.48·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{4} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(1.22304\times 10^{7}\)
Root analytic conductor: \(7.69007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 23^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
23 \( 1 \)
good3$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 4 T + 14 T^{2} + 32 T^{3} + 64 T^{4} + 32 p T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.3.e_o_bg_cm
5$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 2 p T^{2} - 12 T^{3} + 48 T^{4} - 12 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) 4.5.a_k_am_bw
11$D_{4}$ \( ( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) 4.11.i_ce_jo_bmg
13$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 4 T + 40 T^{2} + 124 T^{3} + 694 T^{4} + 124 p T^{5} + 40 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.13.e_bo_eu_bas
17$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 8 T + 82 T^{2} - 412 T^{3} + 2208 T^{4} - 412 p T^{5} + 82 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) 4.17.ai_de_apw_dgy
19$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 28 T^{2} - 88 T^{3} - 794 T^{4} - 88 p T^{5} + 28 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.19.i_bc_adk_abeo
29$D_{4}$ \( ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) 4.29.e_cq_jk_eja
31$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 4 T + 22 T^{2} + 16 T^{3} + 928 T^{4} + 16 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.31.e_w_q_bjs
37$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 16 T + 192 T^{2} - 1616 T^{3} + 11378 T^{4} - 1616 p T^{5} + 192 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) 4.37.aq_hk_acke_qvq
41$D_4\times C_2$ \( 1 + 148 T^{2} + 8790 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} \) 4.41.a_fs_a_nac
43$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 180 T^{2} + 1000 T^{3} + 11750 T^{4} + 1000 p T^{5} + 180 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.43.i_gy_bmm_rjy
47$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 4 T + 110 T^{2} + 400 T^{3} + 7360 T^{4} + 400 p T^{5} + 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) 4.47.e_eg_pk_kxc
53$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 8 T + 104 T^{2} + 776 T^{3} + 6658 T^{4} + 776 p T^{5} + 104 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) 4.53.i_ea_bdw_jwc
59$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 12 T + 166 T^{2} - 888 T^{3} + 9360 T^{4} - 888 p T^{5} + 166 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) 4.59.am_gk_abie_nwa
61$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 24 T + 402 T^{2} + 4476 T^{3} + 40688 T^{4} + 4476 p T^{5} + 402 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) 4.61.y_pm_gqe_ciey
67$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 16 T + 312 T^{2} - 2864 T^{3} + 31298 T^{4} - 2864 p T^{5} + 312 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) 4.67.aq_ma_aege_buhu
71$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 16 T + 148 T^{2} - 16 p T^{3} + 10902 T^{4} - 16 p^{2} T^{5} + 148 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) 4.71.aq_fs_abrs_qdi
73$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 20 T + 360 T^{2} + 4348 T^{3} + 41894 T^{4} + 4348 p T^{5} + 360 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) 4.73.u_nw_glg_cjzi
79$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 24 T + 408 T^{2} - 4680 T^{3} + 47474 T^{4} - 4680 p T^{5} + 408 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) 4.79.ay_ps_agya_csfy
83$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - 28 T + 592 T^{2} - 7772 T^{3} + 84486 T^{4} - 7772 p T^{5} + 592 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) 4.83.abc_wu_almy_euzm
89$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 16 T + 346 T^{2} + 3980 T^{3} + 45840 T^{4} + 3980 p T^{5} + 346 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) 4.89.q_ni_fxc_cpvc
97$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 12 T + 414 T^{2} + 3432 T^{3} + 61448 T^{4} + 3432 p T^{5} + 414 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) 4.97.m_py_fca_dmxk
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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

−5.98094934821140175019636087232, −5.74281897331836959163392383856, −5.72238353054094209786257849499, −5.48402215439293440535806039440, −5.47805246474447533535184293954, −5.15383684779416206225483614193, −4.94860539481649493365705046554, −4.79695267066382271960812184547, −4.69027259654565764327545944227, −4.07874737251560754458104311860, −4.03754143888240257070216028323, −3.81166349297218908400483598015, −3.54367386143652753900528928671, −3.09569193099679465289238643614, −2.99637815843565575504248839448, −2.98674611557305848902358504381, −2.89897668141595992786652967421, −2.33894563062831616323528393918, −2.20389112606291917434149609247, −2.03816846603451166712811732785, −2.02278665593266614091984961822, −1.58139408659043561680374374879, −0.993262842397833889369104992811, −0.78509138224854435201693396120, −0.70798316101581248778635240144, 0, 0, 0, 0, 0.70798316101581248778635240144, 0.78509138224854435201693396120, 0.993262842397833889369104992811, 1.58139408659043561680374374879, 2.02278665593266614091984961822, 2.03816846603451166712811732785, 2.20389112606291917434149609247, 2.33894563062831616323528393918, 2.89897668141595992786652967421, 2.98674611557305848902358504381, 2.99637815843565575504248839448, 3.09569193099679465289238643614, 3.54367386143652753900528928671, 3.81166349297218908400483598015, 4.03754143888240257070216028323, 4.07874737251560754458104311860, 4.69027259654565764327545944227, 4.79695267066382271960812184547, 4.94860539481649493365705046554, 5.15383684779416206225483614193, 5.47805246474447533535184293954, 5.48402215439293440535806039440, 5.72238353054094209786257849499, 5.74281897331836959163392383856, 5.98094934821140175019636087232

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