Properties

Label 8-1232e4-1.1-c3e4-0-0
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $2.79194\times 10^{7}$
Root an. cond. $8.52586$
Motivic weight $3$
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  + 3·3-s − 13·5-s − 28·7-s − 73·9-s − 44·11-s − 34·13-s − 39·15-s − 58·17-s − 60·19-s − 84·21-s + 93·23-s − 175·25-s − 276·27-s − 144·29-s + 129·31-s − 132·33-s + 364·35-s − 187·37-s − 102·39-s + 110·41-s + 360·43-s + 949·45-s + 438·47-s + 490·49-s − 174·51-s + 56·53-s + 572·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.16·5-s − 1.51·7-s − 2.70·9-s − 1.20·11-s − 0.725·13-s − 0.671·15-s − 0.827·17-s − 0.724·19-s − 0.872·21-s + 0.843·23-s − 7/5·25-s − 1.96·27-s − 0.922·29-s + 0.747·31-s − 0.696·33-s + 1.75·35-s − 0.830·37-s − 0.418·39-s + 0.419·41-s + 1.27·43-s + 3.14·45-s + 1.35·47-s + 10/7·49-s − 0.477·51-s + 0.145·53-s + 1.40·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2.79194\times 10^{7}\)
Root analytic conductor: \(8.52586\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.8161149292\)
\(L(\frac12)\) \(\approx\) \(0.8161149292\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
11$C_1$ \( ( 1 + p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - p T + 82 T^{2} - 7 p^{3} T^{3} + 3058 T^{4} - 7 p^{6} T^{5} + 82 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 13 T + 344 T^{2} + 3583 T^{3} + 57262 T^{4} + 3583 p^{3} T^{5} + 344 p^{6} T^{6} + 13 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 34 T + 5606 T^{2} + 276690 T^{3} + 14768138 T^{4} + 276690 p^{3} T^{5} + 5606 p^{6} T^{6} + 34 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 58 T + 13110 T^{2} + 793554 T^{3} + 86309674 T^{4} + 793554 p^{3} T^{5} + 13110 p^{6} T^{6} + 58 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 60 T + 440 p T^{2} - 516 T^{3} + 2996190 T^{4} - 516 p^{3} T^{5} + 440 p^{7} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 93 T + 38136 T^{2} - 2786889 T^{3} + 670366670 T^{4} - 2786889 p^{3} T^{5} + 38136 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 144 T + 45108 T^{2} + 169296 T^{3} + 542137478 T^{4} + 169296 p^{3} T^{5} + 45108 p^{6} T^{6} + 144 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 129 T + 83624 T^{2} - 9927795 T^{3} + 3494444874 T^{4} - 9927795 p^{3} T^{5} + 83624 p^{6} T^{6} - 129 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 187 T + 157682 T^{2} + 21260205 T^{3} + 11096899586 T^{4} + 21260205 p^{3} T^{5} + 157682 p^{6} T^{6} + 187 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 110 T + 128358 T^{2} - 38583654 T^{3} + 8247865066 T^{4} - 38583654 p^{3} T^{5} + 128358 p^{6} T^{6} - 110 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 360 T + 222920 T^{2} - 52617432 T^{3} + 22266915678 T^{4} - 52617432 p^{3} T^{5} + 222920 p^{6} T^{6} - 360 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 438 T + 384570 T^{2} - 120647274 T^{3} + 59266254890 T^{4} - 120647274 p^{3} T^{5} + 384570 p^{6} T^{6} - 438 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 56 T + 215456 T^{2} + 65568472 T^{3} + 16687042222 T^{4} + 65568472 p^{3} T^{5} + 215456 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 1209 T + 1263494 T^{2} - 798055983 T^{3} + 435317853666 T^{4} - 798055983 p^{3} T^{5} + 1263494 p^{6} T^{6} - 1209 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 104 T + 532178 T^{2} + 41162424 T^{3} + 130021700570 T^{4} + 41162424 p^{3} T^{5} + 532178 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 1075 T + 1323752 T^{2} - 799767435 T^{3} + 571821252062 T^{4} - 799767435 p^{3} T^{5} + 1323752 p^{6} T^{6} - 1075 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 963 T + 1736420 T^{2} - 1066540887 T^{3} + 987035994006 T^{4} - 1066540887 p^{3} T^{5} + 1736420 p^{6} T^{6} - 963 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 646 T + 1645142 T^{2} + 749557902 T^{3} + 977273075114 T^{4} + 749557902 p^{3} T^{5} + 1645142 p^{6} T^{6} + 646 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1838 T + 2247664 T^{2} - 1985045486 T^{3} + 1566180762142 T^{4} - 1985045486 p^{3} T^{5} + 2247664 p^{6} T^{6} - 1838 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1238 T + 2320692 T^{2} - 1962315462 T^{3} + 2018945699974 T^{4} - 1962315462 p^{3} T^{5} + 2320692 p^{6} T^{6} - 1238 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1453 T + 3274974 T^{2} + 2929106667 T^{3} + 3558191899714 T^{4} + 2929106667 p^{3} T^{5} + 3274974 p^{6} T^{6} + 1453 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 573 T + 92642 T^{2} + 618376629 T^{3} - 832956043062 T^{4} + 618376629 p^{3} T^{5} + 92642 p^{6} T^{6} - 573 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.70223096581878018545019723737, −6.13454344701040433729659872532, −6.08518604768821151966080712377, −5.86302194675236046796932537823, −5.85344096068421162179684229865, −5.31875903930345737465016923913, −5.28995298558761515829190761369, −5.10493040617229702106225215348, −4.83241792465829420439337476104, −4.29071108640690567179418710344, −4.24578897107977906935667521682, −3.86323829682356233894436801287, −3.80848377392338789084527251378, −3.33125444742264025565610353202, −3.24954139827315818334746568158, −3.14150209122101543032529328726, −2.74813682840099149412317275962, −2.30207858446712056223111415138, −2.27108451410618156875243070857, −2.18284754310610821180115342855, −1.94476638021518614291412281208, −0.76406783676790051928028391155, −0.59314688408736745719628369655, −0.59093956876685816823265618532, −0.18991349167247393819029701601, 0.18991349167247393819029701601, 0.59093956876685816823265618532, 0.59314688408736745719628369655, 0.76406783676790051928028391155, 1.94476638021518614291412281208, 2.18284754310610821180115342855, 2.27108451410618156875243070857, 2.30207858446712056223111415138, 2.74813682840099149412317275962, 3.14150209122101543032529328726, 3.24954139827315818334746568158, 3.33125444742264025565610353202, 3.80848377392338789084527251378, 3.86323829682356233894436801287, 4.24578897107977906935667521682, 4.29071108640690567179418710344, 4.83241792465829420439337476104, 5.10493040617229702106225215348, 5.28995298558761515829190761369, 5.31875903930345737465016923913, 5.85344096068421162179684229865, 5.86302194675236046796932537823, 6.08518604768821151966080712377, 6.13454344701040433729659872532, 6.70223096581878018545019723737

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