Properties

Label 8-2070e4-1.1-c3e4-0-3
Degree $8$
Conductor $1.836\times 10^{13}$
Sign $1$
Analytic cond. $2.22508\times 10^{8}$
Root an. cond. $11.0514$
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  + 8·2-s + 40·4-s + 20·5-s − 7-s + 160·8-s + 160·10-s + 39·11-s − 20·13-s − 8·14-s + 560·16-s + 23·17-s + 53·19-s + 800·20-s + 312·22-s + 92·23-s + 250·25-s − 160·26-s − 40·28-s − 161·29-s + 388·31-s + 1.79e3·32-s + 184·34-s − 20·35-s + 466·37-s + 424·38-s + 3.20e3·40-s − 484·41-s + ⋯
L(s)  = 1  + 2.82·2-s + 5·4-s + 1.78·5-s − 0.0539·7-s + 7.07·8-s + 5.05·10-s + 1.06·11-s − 0.426·13-s − 0.152·14-s + 35/4·16-s + 0.328·17-s + 0.639·19-s + 8.94·20-s + 3.02·22-s + 0.834·23-s + 2·25-s − 1.20·26-s − 0.269·28-s − 1.03·29-s + 2.24·31-s + 9.89·32-s + 0.928·34-s − 0.0965·35-s + 2.07·37-s + 1.81·38-s + 12.6·40-s − 1.84·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(2.22508\times 10^{8}\)
Root analytic conductor: \(11.0514\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2070} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(243.3958106\)
\(L(\frac12)\) \(\approx\) \(243.3958106\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{4} \)
23$C_1$ \( ( 1 - p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 + T - 135 T^{2} + 521 p T^{3} + 125756 T^{4} + 521 p^{4} T^{5} - 135 p^{6} T^{6} + p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 39 T + 323 p T^{2} - 61307 T^{3} + 4937428 T^{4} - 61307 p^{3} T^{5} + 323 p^{7} T^{6} - 39 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 20 T + 3840 T^{2} + 157301 T^{3} + 10278344 T^{4} + 157301 p^{3} T^{5} + 3840 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 23 T + 5447 T^{2} + 251497 T^{3} + 6295268 T^{4} + 251497 p^{3} T^{5} + 5447 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 53 T + 653 p T^{2} - 750501 T^{3} + 76029128 T^{4} - 750501 p^{3} T^{5} + 653 p^{7} T^{6} - 53 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 161 T + 2424 p T^{2} + 9893175 T^{3} + 2240848710 T^{4} + 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} + 161 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 388 T + 132240 T^{2} - 29130015 T^{3} + 5707089166 T^{4} - 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 466 T + 240532 T^{2} - 71494750 T^{3} + 19227960950 T^{4} - 71494750 p^{3} T^{5} + 240532 p^{6} T^{6} - 466 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 484 T + 239452 T^{2} + 82501249 T^{3} + 25012433496 T^{4} + 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} + 484 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 894 T + 484780 T^{2} - 190414350 T^{3} + 59052057270 T^{4} - 190414350 p^{3} T^{5} + 484780 p^{6} T^{6} - 894 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 265 T + 184902 T^{2} - 39015297 T^{3} + 18141867922 T^{4} - 39015297 p^{3} T^{5} + 184902 p^{6} T^{6} - 265 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 576 T + 488156 T^{2} + 177483648 T^{3} + 97785187798 T^{4} + 177483648 p^{3} T^{5} + 488156 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 94 T + 578056 T^{2} - 37319902 T^{3} + 164753482782 T^{4} - 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 1472 T + 1805084 T^{2} + 1370871744 T^{3} + 894579577654 T^{4} + 1370871744 p^{3} T^{5} + 1805084 p^{6} T^{6} + 1472 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 200 T + 364322 T^{2} + 121468265 T^{3} + 278790401066 T^{4} + 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} + 200 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1147 T + 1912188 T^{2} - 1347657401 T^{3} + 1180324718270 T^{4} - 1347657401 p^{3} T^{5} + 1912188 p^{6} T^{6} - 1147 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 908 T + 1048492 T^{2} + 1070426012 T^{3} + 693505308902 T^{4} + 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} + 908 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 1048 T + 1927704 T^{2} - 1122552888 T^{3} + 1371917773086 T^{4} - 1122552888 p^{3} T^{5} + 1927704 p^{6} T^{6} - 1048 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1784 T + 2327792 T^{2} - 1523808488 T^{3} + 1318662054974 T^{4} - 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} - 1784 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 2047 T + 2933893 T^{2} + 2857813609 T^{3} + 3031159147844 T^{4} + 2857813609 p^{3} T^{5} + 2933893 p^{6} T^{6} + 2047 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.20321641463877771912974774381, −5.79399871380085064876019243066, −5.58064050892533669009889014414, −5.53297577015500989333153686960, −5.34644296640410737828141519068, −4.94860856036836182974594273501, −4.88828795539854754750714780675, −4.66025198416688089646774880672, −4.47438992838127563222157657593, −4.23867192627264129939269951892, −3.88793907889407981278967751489, −3.82820270861479506528859621878, −3.68770339208868477079726375899, −3.00382639881459934394182280223, −2.99388554792195916350940846422, −2.86229806421714376891858656027, −2.82994716033130175391670190752, −2.24959710240942508583289448674, −2.11124055181872467287244434321, −1.88918398208748364723503089125, −1.70758839601372981000316093954, −1.17443532375049882395611904096, −1.04067909133511062360647544350, −0.71337618386922101725451085150, −0.59165253085540314681533273102, 0.59165253085540314681533273102, 0.71337618386922101725451085150, 1.04067909133511062360647544350, 1.17443532375049882395611904096, 1.70758839601372981000316093954, 1.88918398208748364723503089125, 2.11124055181872467287244434321, 2.24959710240942508583289448674, 2.82994716033130175391670190752, 2.86229806421714376891858656027, 2.99388554792195916350940846422, 3.00382639881459934394182280223, 3.68770339208868477079726375899, 3.82820270861479506528859621878, 3.88793907889407981278967751489, 4.23867192627264129939269951892, 4.47438992838127563222157657593, 4.66025198416688089646774880672, 4.88828795539854754750714780675, 4.94860856036836182974594273501, 5.34644296640410737828141519068, 5.53297577015500989333153686960, 5.58064050892533669009889014414, 5.79399871380085064876019243066, 6.20321641463877771912974774381

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