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Properties

Label 8-525e4-1.1-c3e4-0-7
Degree $8$
Conductor $75969140625$
Sign $1$
Analytic cond. $920664.$
Root an. cond. $5.56560$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 12·3-s + 10·4-s + 72·6-s − 28·7-s + 3·8-s + 90·9-s + 57·11-s + 120·12-s + 43·13-s − 168·14-s + 54·16-s + 99·17-s + 540·18-s − 12·19-s − 336·21-s + 342·22-s + 156·23-s + 36·24-s + 258·26-s + 540·27-s − 280·28-s + 378·29-s − 93·31-s + 240·32-s + 684·33-s + 594·34-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.30·3-s + 5/4·4-s + 4.89·6-s − 1.51·7-s + 0.132·8-s + 10/3·9-s + 1.56·11-s + 2.88·12-s + 0.917·13-s − 3.20·14-s + 0.843·16-s + 1.41·17-s + 7.07·18-s − 0.144·19-s − 3.49·21-s + 3.31·22-s + 1.41·23-s + 0.306·24-s + 1.94·26-s + 3.84·27-s − 1.88·28-s + 2.42·29-s − 0.538·31-s + 1.32·32-s + 3.60·33-s + 2.99·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(920664.\)
Root analytic conductor: \(5.56560\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(76.23143356\)
\(L(\frac12)\) \(\approx\) \(76.23143356\)
\(L(\frac{5}{2})\) 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)$
bad3$C_1$ \( ( 1 - p T )^{4} \)
5 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - 3 p T + 13 p T^{2} - 99 T^{3} + 149 p T^{4} - 99 p^{3} T^{5} + 13 p^{7} T^{6} - 3 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 57 T + 4001 T^{2} - 182772 T^{3} + 7206874 T^{4} - 182772 p^{3} T^{5} + 4001 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 43 T + 4464 T^{2} - 221093 T^{3} + 13201958 T^{4} - 221093 p^{3} T^{5} + 4464 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 99 T + 13790 T^{2} - 926229 T^{3} + 90687946 T^{4} - 926229 p^{3} T^{5} + 13790 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 12 T - 10212 T^{2} + 154020 T^{3} + 95206646 T^{4} + 154020 p^{3} T^{5} - 10212 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 156 T + 28082 T^{2} - 3613896 T^{3} + 479553835 T^{4} - 3613896 p^{3} T^{5} + 28082 p^{6} T^{6} - 156 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 378 T + 143760 T^{2} - 1025172 p T^{3} + 5852097929 T^{4} - 1025172 p^{4} T^{5} + 143760 p^{6} T^{6} - 378 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 3 p T + 16330 T^{2} - 936891 T^{3} + 385099458 T^{4} - 936891 p^{3} T^{5} + 16330 p^{6} T^{6} + 3 p^{10} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 81 T + 107185 T^{2} - 469746 T^{3} + 5937897606 T^{4} - 469746 p^{3} T^{5} + 107185 p^{6} T^{6} - 81 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 465 T + 105716 T^{2} + 566235 T^{3} - 2050285754 T^{4} + 566235 p^{3} T^{5} + 105716 p^{6} T^{6} + 465 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 64 T + 64662 T^{2} - 2492176 T^{3} + 5477627255 T^{4} - 2492176 p^{3} T^{5} + 64662 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 744 T + 588276 T^{2} - 247094976 T^{3} + 101020317878 T^{4} - 247094976 p^{3} T^{5} + 588276 p^{6} T^{6} - 744 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 729 T + 333446 T^{2} - 137056347 T^{3} + 63585804370 T^{4} - 137056347 p^{3} T^{5} + 333446 p^{6} T^{6} - 729 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 231 T + 701280 T^{2} - 148812447 T^{3} + 204019205798 T^{4} - 148812447 p^{3} T^{5} + 701280 p^{6} T^{6} - 231 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 1353 T + 1374808 T^{2} + 931878711 T^{3} + 511309944510 T^{4} + 931878711 p^{3} T^{5} + 1374808 p^{6} T^{6} + 1353 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 1487 T + 1603371 T^{2} - 1080212116 T^{3} + 674138302124 T^{4} - 1080212116 p^{3} T^{5} + 1603371 p^{6} T^{6} - 1487 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1725 T + 1434819 T^{2} + 706812600 T^{3} + 347044577276 T^{4} + 706812600 p^{3} T^{5} + 1434819 p^{6} T^{6} + 1725 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 512 T + 1030980 T^{2} - 484092568 T^{3} + 566560915526 T^{4} - 484092568 p^{3} T^{5} + 1030980 p^{6} T^{6} - 512 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 1629 T + 1989837 T^{2} - 1837769472 T^{3} + 1504809477470 T^{4} - 1837769472 p^{3} T^{5} + 1989837 p^{6} T^{6} - 1629 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 321 T + 661434 T^{2} - 144132717 T^{3} + 79870144034 T^{4} - 144132717 p^{3} T^{5} + 661434 p^{6} T^{6} - 321 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 978 T + 526228 T^{2} - 159100050 T^{3} - 360762420714 T^{4} - 159100050 p^{3} T^{5} + 526228 p^{6} T^{6} + 978 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 2616 T + 5183100 T^{2} - 6945540936 T^{3} + 7789652954246 T^{4} - 6945540936 p^{3} T^{5} + 5183100 p^{6} T^{6} - 2616 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.21020933624159381362399771471, −7.12130477420466747937687639930, −6.91131837400064379602510785674, −6.72055453061693550594764804643, −6.44088279490559081812844451776, −6.18345146171488203788771783697, −5.76402096855471449556149310098, −5.56806456497069632191265962360, −5.55540386147620267276414724631, −4.77066387449061278807657826576, −4.65955201937060897914681147556, −4.64225306097805184154814645734, −4.38554473540208875189938089407, −3.77649567194735263925024600688, −3.67339395164243619753417093416, −3.60312121316134085675961445744, −3.50238448430169783425923103055, −3.03619470058614862719033602550, −2.86246781913768291286104228164, −2.54186326783310191606004456718, −2.04110872403961636306344438882, −1.71087048605369989933938876335, −1.14647755478358344269672677464, −0.845519908082146173197489452078, −0.73855547433983838570962837547, 0.73855547433983838570962837547, 0.845519908082146173197489452078, 1.14647755478358344269672677464, 1.71087048605369989933938876335, 2.04110872403961636306344438882, 2.54186326783310191606004456718, 2.86246781913768291286104228164, 3.03619470058614862719033602550, 3.50238448430169783425923103055, 3.60312121316134085675961445744, 3.67339395164243619753417093416, 3.77649567194735263925024600688, 4.38554473540208875189938089407, 4.64225306097805184154814645734, 4.65955201937060897914681147556, 4.77066387449061278807657826576, 5.55540386147620267276414724631, 5.56806456497069632191265962360, 5.76402096855471449556149310098, 6.18345146171488203788771783697, 6.44088279490559081812844451776, 6.72055453061693550594764804643, 6.91131837400064379602510785674, 7.12130477420466747937687639930, 7.21020933624159381362399771471

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