Properties

Label 8-55e4-1.1-c2e4-0-1
Degree $8$
Conductor $9150625$
Sign $1$
Analytic cond. $5.04418$
Root an. cond. $1.22419$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 4·3-s + 6·4-s − 5·5-s − 20·6-s + 15·7-s + 10·8-s + 9·9-s + 25·10-s − 11-s + 24·12-s + 5·13-s − 75·14-s − 20·15-s − 10·16-s − 10·17-s − 45·18-s + 50·19-s − 30·20-s + 60·21-s + 5·22-s − 106·23-s + 40·24-s + 10·25-s − 25·26-s + 90·28-s + 90·29-s + ⋯
L(s)  = 1  − 5/2·2-s + 4/3·3-s + 3/2·4-s − 5-s − 3.33·6-s + 15/7·7-s + 5/4·8-s + 9-s + 5/2·10-s − 0.0909·11-s + 2·12-s + 5/13·13-s − 5.35·14-s − 4/3·15-s − 5/8·16-s − 0.588·17-s − 5/2·18-s + 2.63·19-s − 3/2·20-s + 20/7·21-s + 5/22·22-s − 4.60·23-s + 5/3·24-s + 2/5·25-s − 0.961·26-s + 3.21·28-s + 3.10·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9150625 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(9150625\)    =    \(5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.04418\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 9150625,\ (\ :1, 1, 1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4229435137\)
\(L(\frac12)\) \(\approx\) \(0.4229435137\)
\(L(2)\) 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)$
bad5$C_4$ \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 + T - 19 p T^{2} + p^{2} T^{3} + p^{4} T^{4} \)
good2$C_2^2:C_4$ \( 1 + 5 T + 19 T^{2} + 55 T^{3} + 121 T^{4} + 55 p^{2} T^{5} + 19 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \)
3$C_4\times C_2$ \( 1 - 4 T + 7 T^{2} + 8 T^{3} - 95 T^{4} + 8 p^{2} T^{5} + 7 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \)
7$C_2^2:C_4$ \( 1 - 15 T + 109 T^{2} - 225 T^{3} - 44 T^{4} - 225 p^{2} T^{5} + 109 p^{4} T^{6} - 15 p^{6} T^{7} + p^{8} T^{8} \)
13$C_2^2:C_4$ \( 1 - 5 T + 139 T^{2} + 3785 T^{3} - 25664 T^{4} + 3785 p^{2} T^{5} + 139 p^{4} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \)
17$C_2^2:C_4$ \( 1 + 10 T + 189 T^{2} - 6340 T^{3} - 107979 T^{4} - 6340 p^{2} T^{5} + 189 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \)
19$C_2^2:C_4$ \( 1 - 50 T + 1811 T^{2} - 50100 T^{3} + 1054321 T^{4} - 50100 p^{2} T^{5} + 1811 p^{4} T^{6} - 50 p^{6} T^{7} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 53 T + 1759 T^{2} + 53 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 90 T + 3901 T^{2} - 115560 T^{3} + 3213061 T^{4} - 115560 p^{2} T^{5} + 3901 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} \)
31$C_2^2:C_4$ \( 1 - 38 T + 683 T^{2} - 1456 p T^{3} + 2268805 T^{4} - 1456 p^{3} T^{5} + 683 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \)
37$C_2^2:C_4$ \( 1 + 34 T - 513 T^{2} + 20612 T^{3} + 2640605 T^{4} + 20612 p^{2} T^{5} - 513 p^{4} T^{6} + 34 p^{6} T^{7} + p^{8} T^{8} \)
41$C_2^2:C_4$ \( 1 - 85 T + 2871 T^{2} + 1385 p T^{3} - 3624 p^{2} T^{4} + 1385 p^{3} T^{5} + 2871 p^{4} T^{6} - 85 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2:C_4$ \( 1 - 4796 T^{2} + 12580006 T^{4} - 4796 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2:C_4$ \( 1 + 24 T + 3297 T^{2} + 148852 T^{3} + 5349525 T^{4} + 148852 p^{2} T^{5} + 3297 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \)
53$C_2^2:C_4$ \( 1 - 114 T + 4167 T^{2} - 192652 T^{3} + 15249525 T^{4} - 192652 p^{2} T^{5} + 4167 p^{4} T^{6} - 114 p^{6} T^{7} + p^{8} T^{8} \)
59$C_2^2:C_4$ \( 1 + 128 T + 4353 T^{2} - 361484 T^{3} - 44257195 T^{4} - 361484 p^{2} T^{5} + 4353 p^{4} T^{6} + 128 p^{6} T^{7} + p^{8} T^{8} \)
61$C_2^2:C_4$ \( 1 - 140 T + 11841 T^{2} - 800590 T^{3} + 50606061 T^{4} - 800590 p^{2} T^{5} + 11841 p^{4} T^{6} - 140 p^{6} T^{7} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 58 T + 9694 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 - 8 T - 4377 T^{2} - 164506 T^{3} + 25983005 T^{4} - 164506 p^{2} T^{5} - 4377 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \)
73$C_2^2:C_4$ \( 1 + 20 T + 6489 T^{2} + 776470 T^{3} + 10037661 T^{4} + 776470 p^{2} T^{5} + 6489 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \)
79$C_2^2:C_4$ \( 1 + 210 T + 14641 T^{2} - 466410 T^{3} - 121636919 T^{4} - 466410 p^{2} T^{5} + 14641 p^{4} T^{6} + 210 p^{6} T^{7} + p^{8} T^{8} \)
83$C_2^2:C_4$ \( 1 + 410 T + 84789 T^{2} + 11491060 T^{3} + 1115698821 T^{4} + 11491060 p^{2} T^{5} + 84789 p^{4} T^{6} + 410 p^{6} T^{7} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 39 T + 4461 T^{2} + 39 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 - 176 T + 5447 T^{2} - 189598 T^{3} + 79083325 T^{4} - 189598 p^{2} T^{5} + 5447 p^{4} T^{6} - 176 p^{6} T^{7} + p^{8} T^{8} \)
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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

−11.19824423352547144881804523112, −10.55489621726642510941508021024, −10.23625592359672782676046153765, −9.923781588503006806313170308791, −9.873703814619054848950364453909, −9.750599341174255193477456645872, −9.148834504893940431398537766125, −8.722918574176207922991373068733, −8.491283008538359413405562228653, −8.346055344306731124970086317286, −8.237609150389075858165172740842, −8.000137526069435782084187690805, −7.62185293741932143023759372193, −7.43874484079485565933965569736, −6.97907281366596690167182505374, −6.13247294858654957642299270312, −5.67619201958796238102133331354, −5.37144260332235826533317248814, −4.43595976976374641582624211482, −4.38548025344763502211632601462, −4.01807932951661559705896928896, −3.27360523937657420543275860809, −2.52092769498907675760958538011, −1.57389670735529395265651748785, −0.799033656834846895973355624226, 0.799033656834846895973355624226, 1.57389670735529395265651748785, 2.52092769498907675760958538011, 3.27360523937657420543275860809, 4.01807932951661559705896928896, 4.38548025344763502211632601462, 4.43595976976374641582624211482, 5.37144260332235826533317248814, 5.67619201958796238102133331354, 6.13247294858654957642299270312, 6.97907281366596690167182505374, 7.43874484079485565933965569736, 7.62185293741932143023759372193, 8.000137526069435782084187690805, 8.237609150389075858165172740842, 8.346055344306731124970086317286, 8.491283008538359413405562228653, 8.722918574176207922991373068733, 9.148834504893940431398537766125, 9.750599341174255193477456645872, 9.873703814619054848950364453909, 9.923781588503006806313170308791, 10.23625592359672782676046153765, 10.55489621726642510941508021024, 11.19824423352547144881804523112

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