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Properties

Label 8-2e36-1.1-c3e4-0-14
Degree $8$
Conductor $68719476736$
Sign $1$
Analytic cond. $832806.$
Root an. cond. $5.49626$
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 − 12·5-s + 72·9-s − 44·11-s − 20·13-s + 144·15-s − 208·17-s − 116·19-s + 72·25-s − 132·27-s − 220·29-s − 176·31-s + 528·33-s + 348·37-s + 240·39-s + 44·43-s − 864·45-s + 720·47-s − 116·49-s + 2.49e3·51-s − 612·53-s + 528·55-s + 1.39e3·57-s − 1.79e3·59-s − 852·61-s + 240·65-s − 2.29e3·67-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.07·5-s + 8/3·9-s − 1.20·11-s − 0.426·13-s + 2.47·15-s − 2.96·17-s − 1.40·19-s + 0.575·25-s − 0.940·27-s − 1.40·29-s − 1.01·31-s + 2.78·33-s + 1.54·37-s + 0.985·39-s + 0.156·43-s − 2.86·45-s + 2.23·47-s − 0.338·49-s + 6.85·51-s − 1.58·53-s + 1.29·55-s + 3.23·57-s − 3.96·59-s − 1.78·61-s + 0.457·65-s − 4.17·67-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{36}\)
Sign: $1$
Analytic conductor: \(832806.\)
Root analytic conductor: \(5.49626\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{36} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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 \( 1 \)
good3$D_4\times C_2$ \( 1 + 4 p T + 8 p^{2} T^{2} + 44 p T^{3} - 14 T^{4} + 44 p^{4} T^{5} + 8 p^{8} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \)
5$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 84 T^{3} - 13826 T^{4} + 84 p^{3} T^{5} + 72 p^{6} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 + 116 T^{2} + 54790 T^{4} + 116 p^{6} T^{6} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 + 4 p T + 8 p^{2} T^{2} - 372 p T^{3} - 2010478 T^{4} - 372 p^{4} T^{5} + 8 p^{8} T^{6} + 4 p^{10} T^{7} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 + 20 T + 200 T^{2} + 20460 T^{3} + 714782 T^{4} + 20460 p^{3} T^{5} + 200 p^{6} T^{6} + 20 p^{9} T^{7} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 104 T + 12258 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 + 116 T + 6728 T^{2} + 986812 T^{3} + 142022194 T^{4} + 986812 p^{3} T^{5} + 6728 p^{6} T^{6} + 116 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 404 p T^{2} + 314827206 T^{4} - 404 p^{7} T^{6} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 220 T + 24200 T^{2} + 5230500 T^{3} + 1130124254 T^{4} + 5230500 p^{3} T^{5} + 24200 p^{6} T^{6} + 220 p^{9} T^{7} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 + 88 T + 48190 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 348 T + 60552 T^{2} - 14896836 T^{3} + 3603318782 T^{4} - 14896836 p^{3} T^{5} + 60552 p^{6} T^{6} - 348 p^{9} T^{7} + p^{12} T^{8} \)
41$D_4\times C_2$ \( 1 - 186980 T^{2} + 17544268582 T^{4} - 186980 p^{6} T^{6} + p^{12} T^{8} \)
43$D_4\times C_2$ \( 1 - 44 T + 968 T^{2} - 1233540 T^{3} - 1077444334 T^{4} - 1233540 p^{3} T^{5} + 968 p^{6} T^{6} - 44 p^{9} T^{7} + p^{12} T^{8} \)
47$D_{4}$ \( ( 1 - 360 T + 178846 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 + 612 T + 187272 T^{2} + 5820732 T^{3} - 19241963714 T^{4} + 5820732 p^{3} T^{5} + 187272 p^{6} T^{6} + 612 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 + 1796 T + 1612808 T^{2} + 1066082252 T^{3} + 553985601874 T^{4} + 1066082252 p^{3} T^{5} + 1612808 p^{6} T^{6} + 1796 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 + 852 T + 362952 T^{2} + 267799788 T^{3} + 189964941278 T^{4} + 267799788 p^{3} T^{5} + 362952 p^{6} T^{6} + 852 p^{9} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 2292 T + 2626632 T^{2} + 2109545340 T^{3} + 1310310096626 T^{4} + 2109545340 p^{3} T^{5} + 2626632 p^{6} T^{6} + 2292 p^{9} T^{7} + p^{12} T^{8} \)
71$D_4\times C_2$ \( 1 - 1201676 T^{2} + 606649761286 T^{4} - 1201676 p^{6} T^{6} + p^{12} T^{8} \)
73$D_4\times C_2$ \( 1 - 954684 T^{2} + 450988317542 T^{4} - 954684 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 + 1600 T + 1469406 T^{2} + 1600 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 1732 T + 1499912 T^{2} - 1638324780 T^{3} + 1649538614066 T^{4} - 1638324780 p^{3} T^{5} + 1499912 p^{6} T^{6} - 1732 p^{9} T^{7} + p^{12} T^{8} \)
89$D_4\times C_2$ \( 1 - 2607612 T^{2} + 2686461102758 T^{4} - 2607612 p^{6} T^{6} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 280 T + 1342018 T^{2} - 280 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−7.81683873958808486702131893795, −7.47823310513477692420970273803, −7.46696383128152940993539944392, −7.18045359042601440395828085368, −7.04982556531306182910126823372, −6.42112452688283760031392171800, −6.40838653150865764364322433227, −6.15667836276622264292584699499, −6.11704269263160050044226977097, −5.82195617832387378301750108648, −5.40035736095580163058432863011, −5.22750547805922789151214019061, −4.86596167144526638728883830882, −4.51938529416116991440219905724, −4.48802888206486015464234390514, −4.33842378709585552981229622819, −4.11962104000149202409779111487, −3.73371195193730184814931334422, −2.91773939322688774017191775932, −2.88888909621482737791285555668, −2.78840630619006197819309325581, −2.11616973373582944342198417739, −1.64714271363287545649572587763, −1.57621262951771097898828377940, −0.857403817761387428219511232212, 0, 0, 0, 0, 0.857403817761387428219511232212, 1.57621262951771097898828377940, 1.64714271363287545649572587763, 2.11616973373582944342198417739, 2.78840630619006197819309325581, 2.88888909621482737791285555668, 2.91773939322688774017191775932, 3.73371195193730184814931334422, 4.11962104000149202409779111487, 4.33842378709585552981229622819, 4.48802888206486015464234390514, 4.51938529416116991440219905724, 4.86596167144526638728883830882, 5.22750547805922789151214019061, 5.40035736095580163058432863011, 5.82195617832387378301750108648, 6.11704269263160050044226977097, 6.15667836276622264292584699499, 6.40838653150865764364322433227, 6.42112452688283760031392171800, 7.04982556531306182910126823372, 7.18045359042601440395828085368, 7.46696383128152940993539944392, 7.47823310513477692420970273803, 7.81683873958808486702131893795

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