Properties

Label 8-96e8-1.1-c1e4-0-11
Degree $8$
Conductor $7.214\times 10^{15}$
Sign $1$
Analytic cond. $2.93277\times 10^{7}$
Root an. cond. $8.57846$
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  − 8·7-s − 8·13-s + 16·23-s − 8·25-s − 8·31-s − 8·37-s + 16·47-s + 20·49-s + 16·59-s − 24·61-s + 16·67-s + 16·71-s − 8·73-s − 24·79-s + 8·89-s + 64·91-s − 16·97-s − 16·101-s − 24·103-s + 16·107-s − 24·109-s + 8·113-s − 28·121-s − 16·125-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 3.02·7-s − 2.21·13-s + 3.33·23-s − 8/5·25-s − 1.43·31-s − 1.31·37-s + 2.33·47-s + 20/7·49-s + 2.08·59-s − 3.07·61-s + 1.95·67-s + 1.89·71-s − 0.936·73-s − 2.70·79-s + 0.847·89-s + 6.70·91-s − 1.62·97-s − 1.59·101-s − 2.36·103-s + 1.54·107-s − 2.29·109-s + 0.752·113-s − 2.54·121-s − 1.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(2.93277\times 10^{7}\)
Root analytic conductor: \(8.57846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
3 \( 1 \)
good5$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T^{2} + 16 T^{3} + 26 T^{4} + 16 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 44 T^{2} + 24 p T^{3} + 510 T^{4} + 24 p^{2} T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2^2:C_4$ \( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 40 T^{2} + 136 T^{3} + 514 T^{4} + 136 p T^{5} + 40 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 36 T^{2} + 64 T^{3} + 614 T^{4} + 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 44 T^{2} - 64 T^{3} + 918 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 72 T^{2} + 112 T^{3} + 2426 T^{4} + 112 p T^{5} + 72 p^{2} T^{6} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 108 T^{2} + 616 T^{3} + 162 p T^{4} + 616 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 104 T^{2} + 776 T^{3} + 5346 T^{4} + 776 p T^{5} + 104 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 68 T^{2} - 64 T^{3} + 3206 T^{4} - 64 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 76 T^{2} + 64 T^{3} + 3830 T^{4} + 64 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 104 T^{2} + 272 T^{3} + 5594 T^{4} + 272 p T^{5} + 104 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 204 T^{2} - 1552 T^{3} + 12758 T^{4} - 1552 p T^{5} + 204 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 392 T^{2} + 4568 T^{3} + 40386 T^{4} + 4568 p T^{5} + 392 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 172 T^{2} - 912 T^{3} + 6134 T^{4} - 912 p T^{5} + 172 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 4 p T^{2} - 2640 T^{3} + 28070 T^{4} - 2640 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 172 T^{2} + 1720 T^{3} + 16006 T^{4} + 1720 p T^{5} + 172 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 396 T^{2} + 4344 T^{3} + 42398 T^{4} + 4344 p T^{5} + 396 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 252 T^{2} + 128 T^{3} + 28598 T^{4} + 128 p T^{5} + 252 p^{2} T^{6} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 188 T^{2} - 2424 T^{3} + 17894 T^{4} - 2424 p T^{5} + 188 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 356 T^{2} + 4400 T^{3} + 49990 T^{4} + 4400 p T^{5} + 356 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.85951527965114274953070835503, −5.45918503753960497512167745366, −5.28547442647185635614196419538, −5.22281794701898443932555358518, −5.22191299294786727078891119896, −4.99639436980184845848476970269, −4.80507076179087764660089280172, −4.39811412352781098384662575543, −4.27951632518903876949232229456, −3.96476122789539869620402543354, −3.80230715767819366573508633171, −3.73658463048441612020983547773, −3.72608799537740101089371549944, −3.18953753960694329266237274877, −3.06304532186079473337230577163, −3.01909807196143690941309394547, −2.75132115437229098395546682756, −2.60146756728832620842640471177, −2.41178357261953016740689264437, −2.16336351009512694627302586270, −2.10657476061549144593246268333, −1.37437861397914592477609267854, −1.29922943881697012518481396061, −1.23666480672488397365551188686, −0.78822635668050129084102553276, 0, 0, 0, 0, 0.78822635668050129084102553276, 1.23666480672488397365551188686, 1.29922943881697012518481396061, 1.37437861397914592477609267854, 2.10657476061549144593246268333, 2.16336351009512694627302586270, 2.41178357261953016740689264437, 2.60146756728832620842640471177, 2.75132115437229098395546682756, 3.01909807196143690941309394547, 3.06304532186079473337230577163, 3.18953753960694329266237274877, 3.72608799537740101089371549944, 3.73658463048441612020983547773, 3.80230715767819366573508633171, 3.96476122789539869620402543354, 4.27951632518903876949232229456, 4.39811412352781098384662575543, 4.80507076179087764660089280172, 4.99639436980184845848476970269, 5.22191299294786727078891119896, 5.22281794701898443932555358518, 5.28547442647185635614196419538, 5.45918503753960497512167745366, 5.85951527965114274953070835503

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