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Properties

Label 8-230e4-1.1-c3e4-0-1
Degree $8$
Conductor $2798410000$
Sign $1$
Analytic cond. $33913.7$
Root an. cond. $3.68380$
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 + 4·3-s + 40·4-s − 20·5-s + 32·6-s + 26·7-s + 160·8-s − 14·9-s − 160·10-s + 93·11-s + 160·12-s + 32·13-s + 208·14-s − 80·15-s + 560·16-s + 108·17-s − 112·18-s + 185·19-s − 800·20-s + 104·21-s + 744·22-s + 92·23-s + 640·24-s + 250·25-s + 256·26-s − 27·27-s + 1.04e3·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 0.769·3-s + 5·4-s − 1.78·5-s + 2.17·6-s + 1.40·7-s + 7.07·8-s − 0.518·9-s − 5.05·10-s + 2.54·11-s + 3.84·12-s + 0.682·13-s + 3.97·14-s − 1.37·15-s + 35/4·16-s + 1.54·17-s − 1.46·18-s + 2.23·19-s − 8.94·20-s + 1.08·21-s + 7.21·22-s + 0.834·23-s + 5.44·24-s + 2·25-s + 1.93·26-s − 0.192·27-s + 7.01·28-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(33913.7\)
Root analytic conductor: \(3.68380\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(62.45018969\)
\(L(\frac12)\) \(\approx\) \(62.45018969\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
23$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 - 4 T + 10 p T^{2} - 149 T^{3} + 1310 T^{4} - 149 p^{3} T^{5} + 10 p^{7} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 26 T + 724 T^{2} - 10099 T^{3} + 260534 T^{4} - 10099 p^{3} T^{5} + 724 p^{6} T^{6} - 26 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 93 T + 6725 T^{2} - 31627 p T^{3} + 14317128 T^{4} - 31627 p^{4} T^{5} + 6725 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 32 T + 976 T^{2} + 52631 T^{3} - 4682512 T^{4} + 52631 p^{3} T^{5} + 976 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 108 T + 18260 T^{2} - 93099 p T^{3} + 131243742 T^{4} - 93099 p^{4} T^{5} + 18260 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 185 T + 33223 T^{2} - 186623 p T^{3} + 363066172 T^{4} - 186623 p^{4} T^{5} + 33223 p^{6} T^{6} - 185 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 294 T + 42335 T^{2} + 2109052 T^{3} - 859672512 T^{4} + 2109052 p^{3} T^{5} + 42335 p^{6} T^{6} - 294 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 211 T + 103384 T^{2} + 13064236 T^{3} + 4073558761 T^{4} + 13064236 p^{3} T^{5} + 103384 p^{6} T^{6} + 211 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 5 T + 134830 T^{2} + 4948045 T^{3} + 8513448058 T^{4} + 4948045 p^{3} T^{5} + 134830 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 9 p T + 291554 T^{2} + 74201826 T^{3} + 30573978075 T^{4} + 74201826 p^{3} T^{5} + 291554 p^{6} T^{6} + 9 p^{10} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 100 T + 82204 T^{2} + 9571140 T^{3} + 522019158 T^{4} + 9571140 p^{3} T^{5} + 82204 p^{6} T^{6} + 100 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 363 T + 163502 T^{2} - 6031301 T^{3} + 2659565058 T^{4} - 6031301 p^{3} T^{5} + 163502 p^{6} T^{6} + 363 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 21 T + 269018 T^{2} + 89970593 T^{3} + 27668065578 T^{4} + 89970593 p^{3} T^{5} + 269018 p^{6} T^{6} - 21 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 33 T + 343586 T^{2} + 84316061 T^{3} + 56145878106 T^{4} + 84316061 p^{3} T^{5} + 343586 p^{6} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 307 T + 487855 T^{2} + 298124379 T^{3} + 113148939756 T^{4} + 298124379 p^{3} T^{5} + 487855 p^{6} T^{6} + 307 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 725 T + 912826 T^{2} - 351937325 T^{3} + 318440704618 T^{4} - 351937325 p^{3} T^{5} + 912826 p^{6} T^{6} - 725 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 1257 T + 1758566 T^{2} + 1300005098 T^{3} + 998915687661 T^{4} + 1300005098 p^{3} T^{5} + 1758566 p^{6} T^{6} + 1257 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 509 T + 697972 T^{2} - 203398343 T^{3} + 321017582158 T^{4} - 203398343 p^{3} T^{5} + 697972 p^{6} T^{6} - 509 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 1202 T + 1003684 T^{2} - 146546698 T^{3} + 4581734390 T^{4} - 146546698 p^{3} T^{5} + 1003684 p^{6} T^{6} - 1202 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 1377 T + 1324064 T^{2} + 677164645 T^{3} + 435520077246 T^{4} + 677164645 p^{3} T^{5} + 1324064 p^{6} T^{6} + 1377 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 984 T + 2444900 T^{2} + 1775501064 T^{3} + 2545577861478 T^{4} + 1775501064 p^{3} T^{5} + 2444900 p^{6} T^{6} + 984 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 137 T + 867295 T^{2} - 1257176833 T^{3} + 651849046340 T^{4} - 1257176833 p^{3} T^{5} + 867295 p^{6} T^{6} - 137 p^{9} T^{7} + p^{12} T^{8} \)
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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

−8.408571040257319184711404671228, −7.86412058050232159126202766445, −7.84289999927998245593390808055, −7.72764494713986537189133122686, −7.13817456767071836379134482158, −7.11143538519143037905885860852, −6.68167572917697175433594143500, −6.51236104036973715589930832897, −6.39324669800459979715156395795, −5.66306889719802068330073705935, −5.47209119178626565991916149599, −5.35282466702606498697033631619, −5.04514031914002366085434525712, −4.57122697833937927780434391035, −4.36306223199127860417851480160, −4.20802066450498989423814407225, −3.79221102266587888045541856029, −3.34954089972216128614177621414, −3.28142027554712280462588701528, −3.06969447281731746586416734481, −2.89640745608174341720521656897, −1.87860545020693535218493606153, −1.56068778982998718654644880591, −1.19170325940813855819159687912, −0.867854554964773391357946850720, 0.867854554964773391357946850720, 1.19170325940813855819159687912, 1.56068778982998718654644880591, 1.87860545020693535218493606153, 2.89640745608174341720521656897, 3.06969447281731746586416734481, 3.28142027554712280462588701528, 3.34954089972216128614177621414, 3.79221102266587888045541856029, 4.20802066450498989423814407225, 4.36306223199127860417851480160, 4.57122697833937927780434391035, 5.04514031914002366085434525712, 5.35282466702606498697033631619, 5.47209119178626565991916149599, 5.66306889719802068330073705935, 6.39324669800459979715156395795, 6.51236104036973715589930832897, 6.68167572917697175433594143500, 7.11143538519143037905885860852, 7.13817456767071836379134482158, 7.72764494713986537189133122686, 7.84289999927998245593390808055, 7.86412058050232159126202766445, 8.408571040257319184711404671228

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