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Properties

Label 8-1536e4-1.1-c3e4-0-8
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $6.74573\times 10^{7}$
Root an. cond. $9.51981$
Motivic weight $3$
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  + 12·3-s − 8·5-s − 40·7-s + 90·9-s − 40·11-s − 16·13-s − 96·15-s + 24·17-s + 240·19-s − 480·21-s + 48·23-s − 232·25-s + 540·27-s − 360·29-s + 40·31-s − 480·33-s + 320·35-s + 192·37-s − 192·39-s + 392·41-s − 224·43-s − 720·45-s − 848·47-s − 8·49-s + 288·51-s − 1.03e3·53-s + 320·55-s + ⋯
L(s)  = 1  + 2.30·3-s − 0.715·5-s − 2.15·7-s + 10/3·9-s − 1.09·11-s − 0.341·13-s − 1.65·15-s + 0.342·17-s + 2.89·19-s − 4.98·21-s + 0.435·23-s − 1.85·25-s + 3.84·27-s − 2.30·29-s + 0.231·31-s − 2.53·33-s + 1.54·35-s + 0.853·37-s − 0.788·39-s + 1.49·41-s − 0.794·43-s − 2.38·45-s − 2.63·47-s − 0.0233·49-s + 0.790·51-s − 2.67·53-s + 0.784·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(6.74573\times 10^{7}\)
Root analytic conductor: \(9.51981\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 296 T^{2} + 2888 T^{3} + 44738 T^{4} + 2888 p^{3} T^{5} + 296 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 1608 T^{2} + 34936 T^{3} + 808946 T^{4} + 34936 p^{3} T^{5} + 1608 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 3380 T^{2} + 148168 T^{3} + 5752310 T^{4} + 148168 p^{3} T^{5} + 3380 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 3100 T^{2} - 134096 T^{3} + 2018486 T^{4} - 134096 p^{3} T^{5} + 3100 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 4204 T^{2} + 320792 T^{3} + 1340710 T^{4} + 320792 p^{3} T^{5} + 4204 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 240 T + 44124 T^{2} - 5259120 T^{3} + 510477878 T^{4} - 5259120 p^{3} T^{5} + 44124 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 48 T + 44140 T^{2} - 1686896 T^{3} + 780357574 T^{4} - 1686896 p^{3} T^{5} + 44140 p^{6} T^{6} - 48 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 360 T + 99048 T^{2} + 18421992 T^{3} + 3309720418 T^{4} + 18421992 p^{3} T^{5} + 99048 p^{6} T^{6} + 360 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 40 T + 90888 T^{2} - 406456 T^{3} + 3547571282 T^{4} - 406456 p^{3} T^{5} + 90888 p^{6} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 185772 T^{2} - 25679424 T^{3} + 13671262838 T^{4} - 25679424 p^{3} T^{5} + 185772 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 392 T + 269516 T^{2} - 73888568 T^{3} + 27681991942 T^{4} - 73888568 p^{3} T^{5} + 269516 p^{6} T^{6} - 392 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 224 T - 2532 T^{2} + 13935840 T^{3} + 13610249110 T^{4} + 13935840 p^{3} T^{5} - 2532 p^{6} T^{6} + 224 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 848 T + 451596 T^{2} + 177783696 T^{3} + 62501395942 T^{4} + 177783696 p^{3} T^{5} + 451596 p^{6} T^{6} + 848 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 1032 T + 632424 T^{2} + 280529352 T^{3} + 116184917890 T^{4} + 280529352 p^{3} T^{5} + 632424 p^{6} T^{6} + 1032 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 80 T + 447148 T^{2} + 107509200 T^{3} + 104163210230 T^{4} + 107509200 p^{3} T^{5} + 447148 p^{6} T^{6} + 80 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 192 T + 708428 T^{2} - 60508224 T^{3} + 212605389398 T^{4} - 60508224 p^{3} T^{5} + 708428 p^{6} T^{6} - 192 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 288 T + 570668 T^{2} - 32743200 T^{3} + 184484269718 T^{4} - 32743200 p^{3} T^{5} + 570668 p^{6} T^{6} - 288 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1328 T + 1537164 T^{2} + 1039413360 T^{3} + 730992464710 T^{4} + 1039413360 p^{3} T^{5} + 1537164 p^{6} T^{6} + 1328 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 1184 T + 1714404 T^{2} + 1263893216 T^{3} + 1005445776806 T^{4} + 1263893216 p^{3} T^{5} + 1714404 p^{6} T^{6} + 1184 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1320 T + 1537896 T^{2} + 1378961848 T^{3} + 1078534557330 T^{4} + 1378961848 p^{3} T^{5} + 1537896 p^{6} T^{6} + 1320 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 984 T + 1598740 T^{2} + 1220460024 T^{3} + 1229596799574 T^{4} + 1220460024 p^{3} T^{5} + 1598740 p^{6} T^{6} + 984 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1368 T + 1775484 T^{2} + 1068844200 T^{3} + 988506411046 T^{4} + 1068844200 p^{3} T^{5} + 1775484 p^{6} T^{6} + 1368 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 936 T + 1508124 T^{2} + 175463576 T^{3} + 487916627142 T^{4} + 175463576 p^{3} T^{5} + 1508124 p^{6} T^{6} + 936 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.06380817298105305347928216600, −6.50579775478551890857400761199, −6.26926779975163750514006511591, −6.22589726773456838502401278542, −6.19531554400094662015952716779, −5.60212111481907685135755303630, −5.51289754570947887842104269140, −5.14996177948062688382122253515, −5.13727357712070141990999303414, −4.81790379863567484613549017118, −4.33775619489645286029364958109, −4.19165964726733375978105478714, −4.13489925533034767630989975791, −3.55630818825011509273458750039, −3.50939730076784314286019099685, −3.34098119088880442188886223967, −3.25818383946772000166719172065, −2.93661980018405249895257177681, −2.65477651730343959128644633602, −2.57465863073924862064863071191, −2.39164840036093051685121590060, −1.67493170277499661647717651802, −1.44293750062325541205483705024, −1.35383142484583838253992664826, −1.16913599954156120877072244194, 0, 0, 0, 0, 1.16913599954156120877072244194, 1.35383142484583838253992664826, 1.44293750062325541205483705024, 1.67493170277499661647717651802, 2.39164840036093051685121590060, 2.57465863073924862064863071191, 2.65477651730343959128644633602, 2.93661980018405249895257177681, 3.25818383946772000166719172065, 3.34098119088880442188886223967, 3.50939730076784314286019099685, 3.55630818825011509273458750039, 4.13489925533034767630989975791, 4.19165964726733375978105478714, 4.33775619489645286029364958109, 4.81790379863567484613549017118, 5.13727357712070141990999303414, 5.14996177948062688382122253515, 5.51289754570947887842104269140, 5.60212111481907685135755303630, 6.19531554400094662015952716779, 6.22589726773456838502401278542, 6.26926779975163750514006511591, 6.50579775478551890857400761199, 7.06380817298105305347928216600

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