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Properties

Label 8-275e4-1.1-c2e4-0-4
Degree $8$
Conductor $5719140625$
Sign $1$
Analytic cond. $3152.61$
Root an. cond. $2.73737$
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 − 10·3-s + 11·4-s − 50·6-s − 5·7-s + 15·8-s + 49·9-s + 11-s − 110·12-s + 30·13-s − 25·14-s + 10·16-s + 35·17-s + 245·18-s − 50·19-s + 50·21-s + 5·22-s − 50·23-s − 150·24-s + 150·26-s − 180·27-s − 55·28-s + 60·29-s − 43·31-s − 15·32-s − 10·33-s + 175·34-s + ⋯
L(s)  = 1  + 5/2·2-s − 3.33·3-s + 11/4·4-s − 8.33·6-s − 5/7·7-s + 15/8·8-s + 49/9·9-s + 1/11·11-s − 9.16·12-s + 2.30·13-s − 1.78·14-s + 5/8·16-s + 2.05·17-s + 13.6·18-s − 2.63·19-s + 2.38·21-s + 5/22·22-s − 2.17·23-s − 6.25·24-s + 5.76·26-s − 6.66·27-s − 1.96·28-s + 2.06·29-s − 1.38·31-s − 0.468·32-s − 0.303·33-s + 5.14·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3152.61\)
Root analytic conductor: \(2.73737\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 11^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.918097429\)
\(L(\frac12)\) \(\approx\) \(2.918097429\)
\(L(2)\) 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 \( 1 \)
11$C_4$ \( 1 - T - 19 p T^{2} - p^{2} T^{3} + p^{4} T^{4} \)
good2$C_2^2:C_4$ \( 1 - 5 T + 7 p T^{2} - 15 p T^{3} + 61 T^{4} - 15 p^{3} T^{5} + 7 p^{5} T^{6} - 5 p^{6} T^{7} + p^{8} T^{8} \)
3$C_4\times C_2$ \( 1 + 10 T + 17 p T^{2} + 200 T^{3} + 661 T^{4} + 200 p^{2} T^{5} + 17 p^{5} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \)
7$C_2^2:C_4$ \( 1 + 5 T + 34 T^{2} - 545 T^{3} - 3929 T^{4} - 545 p^{2} T^{5} + 34 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \)
13$C_2^2:C_4$ \( 1 - 30 T + 319 T^{2} + 2220 T^{3} - 78839 T^{4} + 2220 p^{2} T^{5} + 319 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \)
17$C_2^2:C_4$ \( 1 - 35 T + 534 T^{2} + 1715 T^{3} - 116889 T^{4} + 1715 p^{2} T^{5} + 534 p^{4} T^{6} - 35 p^{6} T^{7} + p^{8} T^{8} \)
19$C_2^2:C_4$ \( 1 + 50 T + 1461 T^{2} + 34420 T^{3} + 695781 T^{4} + 34420 p^{2} T^{5} + 1461 p^{4} T^{6} + 50 p^{6} T^{7} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 + 25 T + 853 T^{2} + 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 60 T + 1981 T^{2} - 35040 T^{3} + 821401 T^{4} - 35040 p^{2} T^{5} + 1981 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \)
31$C_2^2:C_4$ \( 1 + 43 T + 333 T^{2} - 33259 T^{3} - 1650700 T^{4} - 33259 p^{2} T^{5} + 333 p^{4} T^{6} + 43 p^{6} T^{7} + p^{8} T^{8} \)
37$C_2^2:C_4$ \( 1 + 30 T + 2391 T^{2} + 110740 T^{3} + 2785941 T^{4} + 110740 p^{2} T^{5} + 2391 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
41$C_2^2:C_4$ \( 1 - 95 T + 2726 T^{2} + 115695 T^{3} - 10599329 T^{4} + 115695 p^{2} T^{5} + 2726 p^{4} T^{6} - 95 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2:C_4$ \( 1 - 1266 T^{2} + 5769471 T^{4} - 1266 p^{4} T^{6} + p^{8} T^{8} \)
47$C_2^2:C_4$ \( 1 - 35 T - 1569 T^{2} + 4895 T^{3} + 4987976 T^{4} + 4895 p^{2} T^{5} - 1569 p^{4} T^{6} - 35 p^{6} T^{7} + p^{8} T^{8} \)
53$C_2^2:C_4$ \( 1 - 10 T + 2291 T^{2} - 3920 T^{3} + 5421781 T^{4} - 3920 p^{2} T^{5} + 2291 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \)
59$C_2^2:C_4$ \( 1 - 33 T - 267 T^{2} - 197191 T^{3} + 18592980 T^{4} - 197191 p^{2} T^{5} - 267 p^{4} T^{6} - 33 p^{6} T^{7} + p^{8} T^{8} \)
61$C_2^2:C_4$ \( 1 - 85 T + 10776 T^{2} - 697145 T^{3} + 50099991 T^{4} - 697145 p^{2} T^{5} + 10776 p^{4} T^{6} - 85 p^{6} T^{7} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 170 T + 16123 T^{2} - 170 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2:C_4$ \( 1 + 137 T + 13388 T^{2} + 1284979 T^{3} + 116645215 T^{4} + 1284979 p^{2} T^{5} + 13388 p^{4} T^{6} + 137 p^{6} T^{7} + p^{8} T^{8} \)
73$C_2^2:C_4$ \( 1 + 110 T + 15559 T^{2} + 1501660 T^{3} + 108628801 T^{4} + 1501660 p^{2} T^{5} + 15559 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \)
79$C_2^2:C_4$ \( 1 - 70 T + 4771 T^{2} + 788620 T^{3} - 67577879 T^{4} + 788620 p^{2} T^{5} + 4771 p^{4} T^{6} - 70 p^{6} T^{7} + p^{8} T^{8} \)
83$C_2^2:C_4$ \( 1 + 115 T + 8959 T^{2} - 253135 T^{3} - 27793064 T^{4} - 253135 p^{2} T^{5} + 8959 p^{4} T^{6} + 115 p^{6} T^{7} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 159 T + 21501 T^{2} + 159 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 - 155 T + 23856 T^{2} - 2777845 T^{3} + 359203751 T^{4} - 2777845 p^{2} T^{5} + 23856 p^{4} T^{6} - 155 p^{6} T^{7} + p^{8} T^{8} \)
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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

−8.519735654354739489566156360223, −7.939198268829029476832790263442, −7.938657632020006901287786707758, −7.32342055625197395712658957524, −6.94404643663538203141259424534, −6.92320045976390063563139993496, −6.39579946881156166470901417231, −6.32848952734508977942745227678, −6.07199119843734315927206625843, −5.89634174568419521102945140966, −5.68408765954155112315748753926, −5.56540264470854136631659859956, −5.35519628965900254631252387076, −4.93374736497932833476606878253, −4.49438602323308996145612493370, −4.38979531312384116806025468114, −3.99617083634462295663239218766, −3.90651117020340757594780550901, −3.65066777362257348243694133519, −3.23324196691708604863169496072, −2.56892379118788259572659159989, −2.00537657335147251508904430555, −1.58202439756417858782572512483, −0.65012964774728400070794961290, −0.57829738755818311103609814199, 0.57829738755818311103609814199, 0.65012964774728400070794961290, 1.58202439756417858782572512483, 2.00537657335147251508904430555, 2.56892379118788259572659159989, 3.23324196691708604863169496072, 3.65066777362257348243694133519, 3.90651117020340757594780550901, 3.99617083634462295663239218766, 4.38979531312384116806025468114, 4.49438602323308996145612493370, 4.93374736497932833476606878253, 5.35519628965900254631252387076, 5.56540264470854136631659859956, 5.68408765954155112315748753926, 5.89634174568419521102945140966, 6.07199119843734315927206625843, 6.32848952734508977942745227678, 6.39579946881156166470901417231, 6.92320045976390063563139993496, 6.94404643663538203141259424534, 7.32342055625197395712658957524, 7.938657632020006901287786707758, 7.939198268829029476832790263442, 8.519735654354739489566156360223

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