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Properties

Label 8-725e4-1.1-c5e4-0-0
Degree $8$
Conductor $276281640625$
Sign $1$
Analytic cond. $1.82807\times 10^{8}$
Root an. cond. $10.7832$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 28·3-s − 59·4-s + 208·7-s + 168·8-s − 234·9-s − 124·11-s − 1.65e3·12-s + 460·13-s + 1.55e3·16-s − 184·17-s − 2.39e3·19-s + 5.82e3·21-s + 1.19e3·23-s + 4.70e3·24-s − 1.69e4·27-s − 1.22e4·28-s − 3.36e3·29-s − 1.92e4·31-s − 1.44e4·32-s − 3.47e3·33-s + 1.38e4·36-s + 1.09e4·37-s + 1.28e4·39-s − 1.12e3·41-s + 2.14e4·43-s + 7.31e3·44-s − 2.37e4·47-s + ⋯
L(s)  = 1  + 1.79·3-s − 1.84·4-s + 1.60·7-s + 0.928·8-s − 0.962·9-s − 0.308·11-s − 3.31·12-s + 0.754·13-s + 1.52·16-s − 0.154·17-s − 1.52·19-s + 2.88·21-s + 0.469·23-s + 1.66·24-s − 4.47·27-s − 2.95·28-s − 0.742·29-s − 3.59·31-s − 2.49·32-s − 0.555·33-s + 1.77·36-s + 1.31·37-s + 1.35·39-s − 0.104·41-s + 1.76·43-s + 0.569·44-s − 1.56·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(1.82807\times 10^{8}\)
Root analytic conductor: \(10.7832\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 29^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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)$
bad5 \( 1 \)
29$C_1$ \( ( 1 + p^{2} T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + 59 T^{2} - 21 p^{3} T^{3} + 481 p^{2} T^{4} - 21 p^{8} T^{5} + 59 p^{10} T^{6} + p^{20} T^{8} \)
3$C_2 \wr S_4$ \( 1 - 28 T + 1018 T^{2} - 6032 p T^{3} + 42035 p^{2} T^{4} - 6032 p^{6} T^{5} + 1018 p^{10} T^{6} - 28 p^{15} T^{7} + p^{20} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 208 T + 50020 T^{2} - 6850192 T^{3} + 1014497510 T^{4} - 6850192 p^{5} T^{5} + 50020 p^{10} T^{6} - 208 p^{15} T^{7} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 124 T + 424234 T^{2} - 2589040 T^{3} + 81054467547 T^{4} - 2589040 p^{5} T^{5} + 424234 p^{10} T^{6} + 124 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 460 T + 659346 T^{2} + 118850704 T^{3} + 97850261579 T^{4} + 118850704 p^{5} T^{5} + 659346 p^{10} T^{6} - 460 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 184 T + 5453956 T^{2} + 769013352 T^{3} + 11467152125510 T^{4} + 769013352 p^{5} T^{5} + 5453956 p^{10} T^{6} + 184 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2392 T + 6973884 T^{2} + 12798819480 T^{3} + 24894016673110 T^{4} + 12798819480 p^{5} T^{5} + 6973884 p^{10} T^{6} + 2392 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 1192 T + 18381660 T^{2} - 25053529224 T^{3} + 155801395273446 T^{4} - 25053529224 p^{5} T^{5} + 18381660 p^{10} T^{6} - 1192 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 19212 T + 221298386 T^{2} + 1734150261552 T^{3} + 10610786009786723 T^{4} + 1734150261552 p^{5} T^{5} + 221298386 p^{10} T^{6} + 19212 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 10928 T + 259081044 T^{2} - 2170552279376 T^{3} + 26330320398327638 T^{4} - 2170552279376 p^{5} T^{5} + 259081044 p^{10} T^{6} - 10928 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 1120 T + 124939436 T^{2} + 125715602080 T^{3} + 3124670729197126 T^{4} + 125715602080 p^{5} T^{5} + 124939436 p^{10} T^{6} + 1120 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 21420 T + 435014346 T^{2} - 7359136793872 T^{3} + 89454145102817595 T^{4} - 7359136793872 p^{5} T^{5} + 435014346 p^{10} T^{6} - 21420 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 23772 T + 805388370 T^{2} + 10338335631888 T^{3} + 228527211259419395 T^{4} + 10338335631888 p^{5} T^{5} + 805388370 p^{10} T^{6} + 23772 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 8860 T + 21492794 p T^{2} + 7982732306416 T^{3} + 619291319107778907 T^{4} + 7982732306416 p^{5} T^{5} + 21492794 p^{11} T^{6} + 8860 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 10840 T + 1701897548 T^{2} + 4343203272088 T^{3} + 1367718152106467222 T^{4} + 4343203272088 p^{5} T^{5} + 1701897548 p^{10} T^{6} + 10840 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 49448 T + 2057321812 T^{2} - 78522043066328 T^{3} + 2952566139994453046 T^{4} - 78522043066328 p^{5} T^{5} + 2057321812 p^{10} T^{6} - 49448 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 7840 T + 2717901676 T^{2} - 29569800257440 T^{3} + 3898941673833682358 T^{4} - 29569800257440 p^{5} T^{5} + 2717901676 p^{10} T^{6} - 7840 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 48744 T + 6548351764 T^{2} + 230106406041576 T^{3} + 16930032452217232710 T^{4} + 230106406041576 p^{5} T^{5} + 6548351764 p^{10} T^{6} + 48744 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 74992 T + 7968026628 T^{2} - 382291369814608 T^{3} + 24038082722458524710 T^{4} - 382291369814608 p^{5} T^{5} + 7968026628 p^{10} T^{6} - 74992 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 106076 T + 11710838338 T^{2} + 694262350746992 T^{3} + 47791838682726576947 T^{4} + 694262350746992 p^{5} T^{5} + 11710838338 p^{10} T^{6} + 106076 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 62888 T + 14415496844 T^{2} + 604874915361704 T^{3} + 80649041467245964598 T^{4} + 604874915361704 p^{5} T^{5} + 14415496844 p^{10} T^{6} + 62888 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 107568 T + 18707891628 T^{2} - 1720808168131152 T^{3} + \)\(14\!\cdots\!82\)\( T^{4} - 1720808168131152 p^{5} T^{5} + 18707891628 p^{10} T^{6} - 107568 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 49520 T + 17287734156 T^{2} - 2140548822317840 T^{3} + \)\(13\!\cdots\!58\)\( T^{4} - 2140548822317840 p^{5} T^{5} + 17287734156 p^{10} T^{6} - 49520 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.25829016203512408371540389941, −7.18943929809733559147916217782, −6.79130130744968867734248190191, −6.24547421599642155692489108342, −6.13448074240934148726633683476, −5.99730056868086546160730486841, −5.54038879430530207964787735326, −5.52402887490294758724875151764, −5.13651861236746059898230352340, −4.85803432116581246902454607091, −4.79350561065863311414686151428, −4.69511982179899889870517049435, −4.13049943542488389684218751349, −3.76914094846548004057024537084, −3.74692398104611569731743467722, −3.63564361803997121331456573760, −3.48589683825683559379295166634, −2.77367822981141491444928951200, −2.63249023762075256910718772819, −2.31003280874632510271838211059, −2.23528476474178326454498013217, −1.82864606367945013697924106317, −1.43373176872544598585217631932, −1.24920773931922211368448935562, −1.00484394244342486672503472313, 0, 0, 0, 0, 1.00484394244342486672503472313, 1.24920773931922211368448935562, 1.43373176872544598585217631932, 1.82864606367945013697924106317, 2.23528476474178326454498013217, 2.31003280874632510271838211059, 2.63249023762075256910718772819, 2.77367822981141491444928951200, 3.48589683825683559379295166634, 3.63564361803997121331456573760, 3.74692398104611569731743467722, 3.76914094846548004057024537084, 4.13049943542488389684218751349, 4.69511982179899889870517049435, 4.79350561065863311414686151428, 4.85803432116581246902454607091, 5.13651861236746059898230352340, 5.52402887490294758724875151764, 5.54038879430530207964787735326, 5.99730056868086546160730486841, 6.13448074240934148726633683476, 6.24547421599642155692489108342, 6.79130130744968867734248190191, 7.18943929809733559147916217782, 7.25829016203512408371540389941

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