Properties

Label 8-462e4-1.1-c7e4-0-3
Degree $8$
Conductor $45558341136$
Sign $1$
Analytic cond. $4.33839\times 10^{8}$
Root an. cond. $12.0134$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 32·2-s + 108·3-s + 640·4-s + 238·5-s − 3.45e3·6-s − 1.37e3·7-s − 1.02e4·8-s + 7.29e3·9-s − 7.61e3·10-s − 5.32e3·11-s + 6.91e4·12-s − 1.11e4·13-s + 4.39e4·14-s + 2.57e4·15-s + 1.43e5·16-s + 322·17-s − 2.33e5·18-s − 1.42e3·19-s + 1.52e5·20-s − 1.48e5·21-s + 1.70e5·22-s + 6.02e4·23-s − 1.10e6·24-s − 3.78e4·25-s + 3.56e5·26-s + 3.93e5·27-s − 8.78e5·28-s + ⋯
L(s)  = 1  − 2.82·2-s + 2.30·3-s + 5·4-s + 0.851·5-s − 6.53·6-s − 1.51·7-s − 7.07·8-s + 10/3·9-s − 2.40·10-s − 1.20·11-s + 11.5·12-s − 1.40·13-s + 4.27·14-s + 1.96·15-s + 35/4·16-s + 0.0158·17-s − 9.42·18-s − 0.0477·19-s + 4.25·20-s − 3.49·21-s + 3.41·22-s + 1.03·23-s − 16.3·24-s − 0.484·25-s + 3.97·26-s + 3.84·27-s − 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4.33839\times 10^{8}\)
Root analytic conductor: \(12.0134\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 7/2, 7/2, 7/2, 7/2 ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 + p^{3} T )^{4} \)
3$C_1$ \( ( 1 - p^{3} T )^{4} \)
7$C_1$ \( ( 1 + p^{3} T )^{4} \)
11$C_1$ \( ( 1 + p^{3} T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 238 T + 3781 p^{2} T^{2} - 1584842 p^{2} T^{3} + 63602332 p^{3} T^{4} - 1584842 p^{9} T^{5} + 3781 p^{16} T^{6} - 238 p^{21} T^{7} + p^{28} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 11130 T + 214531697 T^{2} + 1822420476678 T^{3} + 19711258459987812 T^{4} + 1822420476678 p^{7} T^{5} + 214531697 p^{14} T^{6} + 11130 p^{21} T^{7} + p^{28} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 322 T + 1332585668 T^{2} - 839658362302 T^{3} + 756742012712952694 T^{4} - 839658362302 p^{7} T^{5} + 1332585668 p^{14} T^{6} - 322 p^{21} T^{7} + p^{28} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 1428 T + 1054861015 T^{2} - 40517170874568 T^{3} + 425137758508690440 T^{4} - 40517170874568 p^{7} T^{5} + 1054861015 p^{14} T^{6} + 1428 p^{21} T^{7} + p^{28} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 60206 T + 9914358448 T^{2} - 494910887601046 T^{3} + 48740115529527080894 T^{4} - 494910887601046 p^{7} T^{5} + 9914358448 p^{14} T^{6} - 60206 p^{21} T^{7} + p^{28} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 92022 T + 58108332585 T^{2} - 4130170941577170 T^{3} + \)\(14\!\cdots\!16\)\( T^{4} - 4130170941577170 p^{7} T^{5} + 58108332585 p^{14} T^{6} - 92022 p^{21} T^{7} + p^{28} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 45642 T + 46470300664 T^{2} - 4695731186389054 T^{3} + \)\(76\!\cdots\!18\)\( T^{4} - 4695731186389054 p^{7} T^{5} + 46470300664 p^{14} T^{6} + 45642 p^{21} T^{7} + p^{28} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 398406 T + 367619787289 T^{2} - 100555459085460094 T^{3} + \)\(51\!\cdots\!44\)\( T^{4} - 100555459085460094 p^{7} T^{5} + 367619787289 p^{14} T^{6} - 398406 p^{21} T^{7} + p^{28} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 120876 T + 411967498256 T^{2} + 84065305515640180 T^{3} + \)\(10\!\cdots\!34\)\( T^{4} + 84065305515640180 p^{7} T^{5} + 411967498256 p^{14} T^{6} + 120876 p^{21} T^{7} + p^{28} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 116486 T + 970291955392 T^{2} + 99531886357332614 T^{3} + \)\(38\!\cdots\!94\)\( T^{4} + 99531886357332614 p^{7} T^{5} + 970291955392 p^{14} T^{6} + 116486 p^{21} T^{7} + p^{28} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 1110688 T + 1001979933251 T^{2} - 36433987295000140 T^{3} - \)\(94\!\cdots\!04\)\( T^{4} - 36433987295000140 p^{7} T^{5} + 1001979933251 p^{14} T^{6} - 1110688 p^{21} T^{7} + p^{28} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 1842098 T + 4586859701060 T^{2} + 6096374647071067830 T^{3} + \)\(79\!\cdots\!94\)\( T^{4} + 6096374647071067830 p^{7} T^{5} + 4586859701060 p^{14} T^{6} + 1842098 p^{21} T^{7} + p^{28} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 1813816 T + 6163917087575 T^{2} + 7639561029002393184 T^{3} + \)\(16\!\cdots\!68\)\( T^{4} + 7639561029002393184 p^{7} T^{5} + 6163917087575 p^{14} T^{6} + 1813816 p^{21} T^{7} + p^{28} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 4871652 T + 20815628484052 T^{2} + 51547220268662065676 T^{3} + \)\(11\!\cdots\!02\)\( T^{4} + 51547220268662065676 p^{7} T^{5} + 20815628484052 p^{14} T^{6} + 4871652 p^{21} T^{7} + p^{28} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 4720420 T + 26638096611523 T^{2} + 75167628099572930196 T^{3} + \)\(24\!\cdots\!52\)\( T^{4} + 75167628099572930196 p^{7} T^{5} + 26638096611523 p^{14} T^{6} + 4720420 p^{21} T^{7} + p^{28} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 5577096 T + 44488114762364 T^{2} + \)\(15\!\cdots\!00\)\( T^{3} + \)\(63\!\cdots\!82\)\( T^{4} + \)\(15\!\cdots\!00\)\( p^{7} T^{5} + 44488114762364 p^{14} T^{6} + 5577096 p^{21} T^{7} + p^{28} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 6068482 T + 54543768747521 T^{2} + \)\(20\!\cdots\!82\)\( T^{3} + \)\(95\!\cdots\!56\)\( T^{4} + \)\(20\!\cdots\!82\)\( p^{7} T^{5} + 54543768747521 p^{14} T^{6} + 6068482 p^{21} T^{7} + p^{28} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 2483948 T + 51275529366892 T^{2} + 89119985001470945916 T^{3} + \)\(12\!\cdots\!94\)\( T^{4} + 89119985001470945916 p^{7} T^{5} + 51275529366892 p^{14} T^{6} + 2483948 p^{21} T^{7} + p^{28} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3096982 T + 78225560173320 T^{2} + \)\(17\!\cdots\!38\)\( T^{3} + \)\(28\!\cdots\!06\)\( T^{4} + \)\(17\!\cdots\!38\)\( p^{7} T^{5} + 78225560173320 p^{14} T^{6} + 3096982 p^{21} T^{7} + p^{28} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 1681524 T + 110944058525908 T^{2} - \)\(15\!\cdots\!96\)\( T^{3} + \)\(68\!\cdots\!22\)\( T^{4} - \)\(15\!\cdots\!96\)\( p^{7} T^{5} + 110944058525908 p^{14} T^{6} - 1681524 p^{21} T^{7} + p^{28} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 7014058 T + 102729724779796 T^{2} + \)\(63\!\cdots\!14\)\( T^{3} - \)\(23\!\cdots\!42\)\( T^{4} + \)\(63\!\cdots\!14\)\( p^{7} T^{5} + 102729724779796 p^{14} T^{6} - 7014058 p^{21} T^{7} + p^{28} T^{8} \)
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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

−7.42067258917107492809684384103, −7.28891797140909571529633399495, −7.02089455890602037949826048310, −6.77724765766270432774603872268, −6.70623502599555238937812204894, −6.00263685969113023834883039063, −5.92040856735250698786596704964, −5.83027292892858635464630130692, −5.80253180781567300033220850684, −4.90482693060292819203954395881, −4.63640643567706822892930949530, −4.57181341669114145264898904626, −4.33357826703743571334376465947, −3.50738491236845005797609490255, −3.29003133404862629952066983235, −3.26462199633379513548086564308, −3.07412748666656443190725955038, −2.50399549850464234912180512625, −2.48798209839648495296190992139, −2.32576114694268588313619564266, −2.24606893730516270469439078822, −1.54613191785396142429436063899, −1.34792040946941637973292304285, −1.13742980721373335198244654343, −1.04615324315521098512336581424, 0, 0, 0, 0, 1.04615324315521098512336581424, 1.13742980721373335198244654343, 1.34792040946941637973292304285, 1.54613191785396142429436063899, 2.24606893730516270469439078822, 2.32576114694268588313619564266, 2.48798209839648495296190992139, 2.50399549850464234912180512625, 3.07412748666656443190725955038, 3.26462199633379513548086564308, 3.29003133404862629952066983235, 3.50738491236845005797609490255, 4.33357826703743571334376465947, 4.57181341669114145264898904626, 4.63640643567706822892930949530, 4.90482693060292819203954395881, 5.80253180781567300033220850684, 5.83027292892858635464630130692, 5.92040856735250698786596704964, 6.00263685969113023834883039063, 6.70623502599555238937812204894, 6.77724765766270432774603872268, 7.02089455890602037949826048310, 7.28891797140909571529633399495, 7.42067258917107492809684384103

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