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Properties

Label 8-44e8-1.1-c3e4-0-3
Degree $8$
Conductor $1.405\times 10^{13}$
Sign $1$
Analytic cond. $1.70249\times 10^{8}$
Root an. cond. $10.6877$
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  + 3·3-s + 9·5-s + 8·7-s − 34·9-s + 80·13-s + 27·15-s − 16·17-s + 8·19-s + 24·21-s + 71·23-s − 204·25-s − 111·27-s + 240·29-s + 115·31-s + 72·35-s + 315·37-s + 240·39-s − 592·41-s − 624·43-s − 306·45-s + 304·47-s − 636·49-s − 48·51-s + 184·53-s + 24·57-s + 805·59-s + 240·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.804·5-s + 0.431·7-s − 1.25·9-s + 1.70·13-s + 0.464·15-s − 0.228·17-s + 0.0965·19-s + 0.249·21-s + 0.643·23-s − 1.63·25-s − 0.791·27-s + 1.53·29-s + 0.666·31-s + 0.347·35-s + 1.39·37-s + 0.985·39-s − 2.25·41-s − 2.21·43-s − 1.01·45-s + 0.943·47-s − 1.85·49-s − 0.131·51-s + 0.476·53-s + 0.0557·57-s + 1.77·59-s + 0.503·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.70249\times 10^{8}\)
Root analytic conductor: \(10.6877\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(13.83940755\)
\(L(\frac12)\) \(\approx\) \(13.83940755\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 \)
good3$D_4\times C_2$ \( 1 - p T + 43 T^{2} - 40 p T^{3} + 952 T^{4} - 40 p^{4} T^{5} + 43 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 57 p T^{2} - 18 p^{3} T^{3} + 44474 T^{4} - 18 p^{6} T^{5} + 57 p^{7} T^{6} - 9 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 100 p T^{2} - 5480 T^{3} + 261286 T^{4} - 5480 p^{3} T^{5} + 100 p^{7} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 80 T + 5012 T^{2} - 232944 T^{3} + 10713302 T^{4} - 232944 p^{3} T^{5} + 5012 p^{6} T^{6} - 80 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 132 p T^{2} - 141840 T^{3} - 12075962 T^{4} - 141840 p^{3} T^{5} + 132 p^{7} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 644 p T^{2} - 624840 T^{3} + 72545174 T^{4} - 624840 p^{3} T^{5} + 644 p^{7} T^{6} - 8 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 71 T + 18567 T^{2} - 845892 T^{3} + 133046236 T^{4} - 845892 p^{3} T^{5} + 18567 p^{6} T^{6} - 71 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 240 T + 57108 T^{2} - 10978320 T^{3} + 2268112790 T^{4} - 10978320 p^{3} T^{5} + 57108 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 115 T + 105407 T^{2} - 9212484 T^{3} + 4535179628 T^{4} - 9212484 p^{3} T^{5} + 105407 p^{6} T^{6} - 115 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 315 T + 169661 T^{2} - 37455894 T^{3} + 11554722546 T^{4} - 37455894 p^{3} T^{5} + 169661 p^{6} T^{6} - 315 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 592 T + 323556 T^{2} + 110265648 T^{3} + 35310327334 T^{4} + 110265648 p^{3} T^{5} + 323556 p^{6} T^{6} + 592 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 624 T + 233420 T^{2} + 33706224 T^{3} + 6261015414 T^{4} + 33706224 p^{3} T^{5} + 233420 p^{6} T^{6} + 624 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 304 T + 283164 T^{2} - 60941424 T^{3} + 35448270982 T^{4} - 60941424 p^{3} T^{5} + 283164 p^{6} T^{6} - 304 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 184 T + 337356 T^{2} - 42426600 T^{3} + 66289132918 T^{4} - 42426600 p^{3} T^{5} + 337356 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 805 T + 714963 T^{2} - 372263808 T^{3} + 212745781840 T^{4} - 372263808 p^{3} T^{5} + 714963 p^{6} T^{6} - 805 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 240 T + 620948 T^{2} - 101342736 T^{3} + 189813668886 T^{4} - 101342736 p^{3} T^{5} + 620948 p^{6} T^{6} - 240 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 119 T + 410251 T^{2} - 89693968 T^{3} + 54926587264 T^{4} - 89693968 p^{3} T^{5} + 410251 p^{6} T^{6} + 119 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 723 T + 716087 T^{2} + 333059148 T^{3} + 209360674884 T^{4} + 333059148 p^{3} T^{5} + 716087 p^{6} T^{6} + 723 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 1936 T + 2403172 T^{2} - 2066847856 T^{3} + 1437251621542 T^{4} - 2066847856 p^{3} T^{5} + 2403172 p^{6} T^{6} - 1936 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 1520 T + 2339644 T^{2} + 1981998896 T^{3} + 1738068432838 T^{4} + 1981998896 p^{3} T^{5} + 2339644 p^{6} T^{6} + 1520 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 2016 T + 2302508 T^{2} + 1947038688 T^{3} + 1505423151318 T^{4} + 1947038688 p^{3} T^{5} + 2302508 p^{6} T^{6} + 2016 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 367 T + 1208649 T^{2} + 193999014 T^{3} + 735902470534 T^{4} + 193999014 p^{3} T^{5} + 1208649 p^{6} T^{6} - 367 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 881 T + 3000353 T^{2} - 16738398 p T^{3} + 3632193512894 T^{4} - 16738398 p^{4} T^{5} + 3000353 p^{6} T^{6} - 881 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.15016647411407411688069195215, −6.04689580350422217995555433288, −5.81126634287880056851680197690, −5.56603833236024441559832085819, −5.40918412361678462786482493829, −4.89155741562359943530884160510, −4.88805687086663777324467341164, −4.86040903187815745808568279288, −4.58259700306519090260705520726, −4.10867442556718353904729159210, −3.83699505987827338386452870290, −3.75230257306213727004099773443, −3.49056897141785859239085672165, −3.23736623256501974966790300329, −3.00657102142221199965480081861, −2.83884651531026412947464969433, −2.53643749057605368574354615048, −2.22089621969681255501833743299, −1.97014159906576482131423808209, −1.77667559830064815337677212028, −1.39059176638105958226895444500, −1.35818000279730136806613145295, −0.68342935842551179741360955038, −0.53218213468958088261893508870, −0.41596623174600211475425384333, 0.41596623174600211475425384333, 0.53218213468958088261893508870, 0.68342935842551179741360955038, 1.35818000279730136806613145295, 1.39059176638105958226895444500, 1.77667559830064815337677212028, 1.97014159906576482131423808209, 2.22089621969681255501833743299, 2.53643749057605368574354615048, 2.83884651531026412947464969433, 3.00657102142221199965480081861, 3.23736623256501974966790300329, 3.49056897141785859239085672165, 3.75230257306213727004099773443, 3.83699505987827338386452870290, 4.10867442556718353904729159210, 4.58259700306519090260705520726, 4.86040903187815745808568279288, 4.88805687086663777324467341164, 4.89155741562359943530884160510, 5.40918412361678462786482493829, 5.56603833236024441559832085819, 5.81126634287880056851680197690, 6.04689580350422217995555433288, 6.15016647411407411688069195215

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