Properties

Label 8-245e4-1.1-c1e4-0-11
Degree $8$
Conductor $3603000625$
Sign $1$
Analytic cond. $14.6478$
Root an. cond. $1.39869$
Motivic weight $1$
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  + 2·2-s + 4·3-s − 4-s − 4·5-s + 8·6-s − 4·8-s + 11·9-s − 8·10-s + 2·11-s − 4·12-s + 8·13-s − 16·15-s − 8·17-s + 22·18-s + 2·19-s + 4·20-s + 4·22-s + 14·23-s − 16·24-s + 5·25-s + 16·26-s + 20·27-s − 32·30-s + 12·31-s + 2·32-s + 8·33-s − 16·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s − 1/2·4-s − 1.78·5-s + 3.26·6-s − 1.41·8-s + 11/3·9-s − 2.52·10-s + 0.603·11-s − 1.15·12-s + 2.21·13-s − 4.13·15-s − 1.94·17-s + 5.18·18-s + 0.458·19-s + 0.894·20-s + 0.852·22-s + 2.91·23-s − 3.26·24-s + 25-s + 3.13·26-s + 3.84·27-s − 5.84·30-s + 2.15·31-s + 0.353·32-s + 1.39·33-s − 2.74·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(14.6478\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(5.416763130\)
\(L(\frac12)\) \(\approx\) \(5.416763130\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7 \( 1 \)
good2$D_4\times C_2$ \( 1 - p T + 5 T^{2} - p^{3} T^{3} + 13 T^{4} - p^{4} T^{5} + 5 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
3$D_4\times C_2$ \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 + 8 T + 20 T^{2} - 52 T^{3} - 545 T^{4} - 52 p T^{5} + 20 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 2 T - 32 T^{2} + 4 T^{3} + 859 T^{4} + 4 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 14 T + 53 T^{2} + 226 T^{3} - 2552 T^{4} + 226 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^3$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 983 T^{4} + 288 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$D_4\times C_2$ \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 6 T + 18 T^{2} + 60 T^{3} - 889 T^{4} + 60 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 + 6 T + 90 T^{2} + 672 T^{3} + 5159 T^{4} + 672 p T^{5} + 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
59$D_4\times C_2$ \( 1 - 6 T - 64 T^{2} + 108 T^{3} + 4395 T^{4} + 108 p T^{5} - 64 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 12 T + 157 T^{2} - 1308 T^{3} + 11088 T^{4} - 1308 p T^{5} + 157 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 8 T + 137 T^{2} - 1224 T^{3} + 11492 T^{4} - 1224 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 144 T^{2} - 600 T^{3} + 10991 T^{4} - 600 p T^{5} + 144 p^{2} T^{6} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 2 T + 2 T^{2} + 140 T^{3} + 9631 T^{4} + 140 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$$\times$$C_2^2$ \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} ) \)
97$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.949612875179780257974502966109, −8.402639803925009913688642744080, −8.352470731670388017030451170634, −8.257376510738481310199078478030, −8.050188143434035454381855578629, −7.47445748740536235799365581415, −7.23880411338807983794650056660, −7.03993690360334365223879399408, −6.66197633749224826987154892839, −6.62611938972739241586156932095, −6.34430712246511704204514345142, −5.56843992951423429966437739872, −5.30417541739839686574351533458, −4.99081220727693525348070557681, −4.68611151283809022353215390164, −4.52847395552872949655662830893, −3.98184147530160152114854506620, −3.81980212625441997279938671993, −3.80868590614437506270269041293, −3.67391235720223874429829754519, −2.95724048426322936430357338095, −2.90988571243839721849702450456, −2.26629895215786835342617923308, −1.50465327764141034868168190830, −1.03753174663562713164984530870, 1.03753174663562713164984530870, 1.50465327764141034868168190830, 2.26629895215786835342617923308, 2.90988571243839721849702450456, 2.95724048426322936430357338095, 3.67391235720223874429829754519, 3.80868590614437506270269041293, 3.81980212625441997279938671993, 3.98184147530160152114854506620, 4.52847395552872949655662830893, 4.68611151283809022353215390164, 4.99081220727693525348070557681, 5.30417541739839686574351533458, 5.56843992951423429966437739872, 6.34430712246511704204514345142, 6.62611938972739241586156932095, 6.66197633749224826987154892839, 7.03993690360334365223879399408, 7.23880411338807983794650056660, 7.47445748740536235799365581415, 8.050188143434035454381855578629, 8.257376510738481310199078478030, 8.352470731670388017030451170634, 8.402639803925009913688642744080, 8.949612875179780257974502966109

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