Properties

Label 8-210e4-1.1-c5e4-0-0
Degree $8$
Conductor $1944810000$
Sign $1$
Analytic cond. $1.28682\times 10^{6}$
Root an. cond. $5.80349$
Motivic weight $5$
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 + 18·3-s + 16·4-s + 50·5-s − 144·6-s − 336·7-s + 128·8-s + 81·9-s − 400·10-s + 482·11-s + 288·12-s − 56·13-s + 2.68e3·14-s + 900·15-s − 1.02e3·16-s − 938·17-s − 648·18-s − 2.19e3·19-s + 800·20-s − 6.04e3·21-s − 3.85e3·22-s − 618·23-s + 2.30e3·24-s + 625·25-s + 448·26-s − 1.45e3·27-s − 5.37e3·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s − 2.59·7-s + 0.707·8-s + 1/3·9-s − 1.26·10-s + 1.20·11-s + 0.577·12-s − 0.0919·13-s + 3.66·14-s + 1.03·15-s − 16-s − 0.787·17-s − 0.471·18-s − 1.39·19-s + 0.447·20-s − 2.99·21-s − 1.69·22-s − 0.243·23-s + 0.816·24-s + 1/5·25-s + 0.129·26-s − 0.384·27-s − 1.29·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1.28682\times 10^{6}\)
Root analytic conductor: \(5.80349\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.01420234719\)
\(L(\frac12)\) \(\approx\) \(0.01420234719\)
\(L(\frac{7}{2})\) 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)$
bad2$C_2$ \( ( 1 + p^{2} T + p^{4} T^{2} )^{2} \)
3$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \)
7$C_2^2$ \( 1 + 48 p T + 1111 p^{2} T^{2} + 48 p^{6} T^{3} + p^{10} T^{4} \)
good11$D_4\times C_2$ \( 1 - 482 T - 129588 T^{2} - 19188420 T^{3} + 67626858619 T^{4} - 19188420 p^{5} T^{5} - 129588 p^{10} T^{6} - 482 p^{15} T^{7} + p^{20} T^{8} \)
13$D_{4}$ \( ( 1 + 28 T + 557807 T^{2} + 28 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 938 T - 385800 T^{2} - 1476477660 T^{3} - 1663391828021 T^{4} - 1476477660 p^{5} T^{5} - 385800 p^{10} T^{6} + 938 p^{15} T^{7} + p^{20} T^{8} \)
19$D_4\times C_2$ \( 1 + 2198 T - 630571 T^{2} + 1120050246 T^{3} + 11737380499628 T^{4} + 1120050246 p^{5} T^{5} - 630571 p^{10} T^{6} + 2198 p^{15} T^{7} + p^{20} T^{8} \)
23$D_4\times C_2$ \( 1 + 618 T + 7974068 T^{2} - 12647264940 T^{3} + 13755945439155 T^{4} - 12647264940 p^{5} T^{5} + 7974068 p^{10} T^{6} + 618 p^{15} T^{7} + p^{20} T^{8} \)
29$D_{4}$ \( ( 1 + 10086 T + 61397308 T^{2} + 10086 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 4138 T - 384419 T^{2} - 164488971782 T^{3} - 1003335238540076 T^{4} - 164488971782 p^{5} T^{5} - 384419 p^{10} T^{6} + 4138 p^{15} T^{7} + p^{20} T^{8} \)
37$D_4\times C_2$ \( 1 - 56 p T - 132465995 T^{2} + 108009160 p T^{3} + 13596782301647104 T^{4} + 108009160 p^{6} T^{5} - 132465995 p^{10} T^{6} - 56 p^{16} T^{7} + p^{20} T^{8} \)
41$D_{4}$ \( ( 1 - 2546 T + 144501292 T^{2} - 2546 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 26700 T + 465893611 T^{2} + 26700 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3484 T - 441176226 T^{2} + 18728353488 T^{3} + 150111038947810483 T^{4} + 18728353488 p^{5} T^{5} - 441176226 p^{10} T^{6} - 3484 p^{15} T^{7} + p^{20} T^{8} \)
53$D_4\times C_2$ \( 1 + 316 T + 195466302 T^{2} - 326035348512 T^{3} - 136761181471991141 T^{4} - 326035348512 p^{5} T^{5} + 195466302 p^{10} T^{6} + 316 p^{15} T^{7} + p^{20} T^{8} \)
59$D_4\times C_2$ \( 1 + 66202 T + 1868766084 T^{2} + 71768934256644 T^{3} + 2675468974890964603 T^{4} + 71768934256644 p^{5} T^{5} + 1868766084 p^{10} T^{6} + 66202 p^{15} T^{7} + p^{20} T^{8} \)
61$D_4\times C_2$ \( 1 - 44472 T + 163486982 T^{2} - 5562522182400 T^{3} + 907831506632786619 T^{4} - 5562522182400 p^{5} T^{5} + 163486982 p^{10} T^{6} - 44472 p^{15} T^{7} + p^{20} T^{8} \)
67$D_4\times C_2$ \( 1 - 18964 T - 2007886963 T^{2} + 6309890866620 T^{3} + 3296764429980902960 T^{4} + 6309890866620 p^{5} T^{5} - 2007886963 p^{10} T^{6} - 18964 p^{15} T^{7} + p^{20} T^{8} \)
71$D_{4}$ \( ( 1 + 34438 T + 897278944 T^{2} + 34438 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 34516 T - 786845663 T^{2} - 74828729803772 T^{3} - 2854113575485200896 T^{4} - 74828729803772 p^{5} T^{5} - 786845663 p^{10} T^{6} + 34516 p^{15} T^{7} + p^{20} T^{8} \)
79$D_4\times C_2$ \( 1 - 121590 T + 4949061533 T^{2} - 447567168772710 T^{3} + 41768662320953606388 T^{4} - 447567168772710 p^{5} T^{5} + 4949061533 p^{10} T^{6} - 121590 p^{15} T^{7} + p^{20} T^{8} \)
83$D_{4}$ \( ( 1 + 110794 T + 8461562296 T^{2} + 110794 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 94246 T + 325861488 T^{2} - 246139627060020 T^{3} - 7568272984699118501 T^{4} - 246139627060020 p^{5} T^{5} + 325861488 p^{10} T^{6} + 94246 p^{15} T^{7} + p^{20} T^{8} \)
97$D_{4}$ \( ( 1 - 71752 T + 18460006626 T^{2} - 71752 p^{5} T^{3} + p^{10} T^{4} )^{2} \)
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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

−8.403830648638118308327261598286, −8.071510539981663954862104962731, −7.59550978786611421825868479288, −7.22677546892351465845304094055, −7.07393256102878243325528115893, −6.84384781519450086954504104914, −6.75233887447594351671998781496, −6.17109246052970509643285717742, −5.96575358871704060355586465922, −5.90545023036167703583265188994, −5.51387837433213202674313189511, −5.01311689096808007965365737469, −4.64872164112814007966150500519, −4.12679678188131471212762581947, −3.79443352287129430532508538398, −3.60027544279768995254150874579, −3.57320132135709584020358518128, −2.81629656119590376384862427592, −2.74543852939687236527382275007, −2.11563153641465900457639889221, −1.85482996052398075882624582190, −1.55852815308144100889312999304, −1.27731307688945001351646522983, −0.16693286097785041014254497589, −0.07475782184884673660145682690, 0.07475782184884673660145682690, 0.16693286097785041014254497589, 1.27731307688945001351646522983, 1.55852815308144100889312999304, 1.85482996052398075882624582190, 2.11563153641465900457639889221, 2.74543852939687236527382275007, 2.81629656119590376384862427592, 3.57320132135709584020358518128, 3.60027544279768995254150874579, 3.79443352287129430532508538398, 4.12679678188131471212762581947, 4.64872164112814007966150500519, 5.01311689096808007965365737469, 5.51387837433213202674313189511, 5.90545023036167703583265188994, 5.96575358871704060355586465922, 6.17109246052970509643285717742, 6.75233887447594351671998781496, 6.84384781519450086954504104914, 7.07393256102878243325528115893, 7.22677546892351465845304094055, 7.59550978786611421825868479288, 8.071510539981663954862104962731, 8.403830648638118308327261598286

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