Properties

Label 8-7406e4-1.1-c1e4-0-0
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 $0$

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: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 7^{4} \cdot 23^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3679086617\)
\(L(\frac12)\) \(\approx\) \(0.3679086617\)
\(L(\frac{3}{2})\) 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_m_bw
11$D_{4}$ \( ( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) 4.11.ai_ce_ajo_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.i_de_pw_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.ai_bc_dk_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.q_hk_cke_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.ai_gy_abmm_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.ai_ea_abdw_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.ay_pm_agqe_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.q_ma_ege_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.y_ps_gya_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.bc_wu_lmy_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.aq_ni_afxc_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.am_py_afca_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

−5.81950034703144943367483499889, −5.37699523891282678415502987681, −5.34009848269080684883934670267, −5.28910271159041393192005745339, −5.04919771334962511746652656153, −4.58733103726191877535396676705, −4.38692854187767968220084398827, −4.36968865103297591211000992205, −4.10374627515662383399524281929, −3.88052226328641630475231885263, −3.48183827892384674300891734951, −3.38195877159965462114251309808, −3.32958753334234216614096668698, −2.73658117638099969419716150191, −2.57627438584882471060495971775, −2.49443631604722549845412402310, −2.17514699342894987220361710208, −1.76416591331940047382495660993, −1.64344930930499182891497901134, −1.64024581454769373799358597589, −1.47279773319997229184653979207, −0.801016731639091983875451887299, −0.68894945982556902778339605276, −0.39540345702300251612584382025, −0.33403602960027873493651292670, 0.33403602960027873493651292670, 0.39540345702300251612584382025, 0.68894945982556902778339605276, 0.801016731639091983875451887299, 1.47279773319997229184653979207, 1.64024581454769373799358597589, 1.64344930930499182891497901134, 1.76416591331940047382495660993, 2.17514699342894987220361710208, 2.49443631604722549845412402310, 2.57627438584882471060495971775, 2.73658117638099969419716150191, 3.32958753334234216614096668698, 3.38195877159965462114251309808, 3.48183827892384674300891734951, 3.88052226328641630475231885263, 4.10374627515662383399524281929, 4.36968865103297591211000992205, 4.38692854187767968220084398827, 4.58733103726191877535396676705, 5.04919771334962511746652656153, 5.28910271159041393192005745339, 5.34009848269080684883934670267, 5.37699523891282678415502987681, 5.81950034703144943367483499889

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