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Properties

Label 8-208e4-1.1-c2e4-0-4
Degree $8$
Conductor $1871773696$
Sign $1$
Analytic cond. $1031.79$
Root an. cond. $2.38066$
Motivic weight $2$
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  + 2·3-s − 14·5-s − 16·7-s + 16·9-s − 4·11-s − 26·13-s − 28·15-s − 12·17-s − 10·19-s − 32·21-s − 18·23-s + 98·25-s + 52·27-s + 2·29-s + 20·31-s − 8·33-s + 224·35-s − 68·37-s − 52·39-s + 100·41-s − 180·43-s − 224·45-s + 68·47-s + 92·49-s − 24·51-s + 128·53-s + 56·55-s + ⋯
L(s)  = 1  + 2/3·3-s − 2.79·5-s − 2.28·7-s + 16/9·9-s − 0.363·11-s − 2·13-s − 1.86·15-s − 0.705·17-s − 0.526·19-s − 1.52·21-s − 0.782·23-s + 3.91·25-s + 1.92·27-s + 2/29·29-s + 0.645·31-s − 0.242·33-s + 32/5·35-s − 1.83·37-s − 4/3·39-s + 2.43·41-s − 4.18·43-s − 4.97·45-s + 1.44·47-s + 1.87·49-s − 0.470·51-s + 2.41·53-s + 1.01·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1031.79\)
Root analytic conductor: \(2.38066\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 13^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7044328192\)
\(L(\frac12)\) \(\approx\) \(0.7044328192\)
\(L(2)\) 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)$
bad2 \( 1 \)
13$C_2$ \( ( 1 + p T + p^{2} T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 2 T - 4 p T^{2} + 4 T^{3} + 139 T^{4} + 4 p^{2} T^{5} - 4 p^{5} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \)
5$C_2^2$$\times$$C_2^2$ \( ( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \)
7$D_4\times C_2$ \( 1 + 16 T + 164 T^{2} + 1236 T^{3} + 8927 T^{4} + 1236 p^{2} T^{5} + 164 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \)
11$D_4\times C_2$ \( 1 + 4 T + 200 T^{2} - 960 T^{3} + 17471 T^{4} - 960 p^{2} T^{5} + 200 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 + 12 T + 413 T^{2} + 4380 T^{3} + 63576 T^{4} + 4380 p^{2} T^{5} + 413 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 10 T + 74 T^{2} + 1752 T^{3} - 96625 T^{4} + 1752 p^{2} T^{5} + 74 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 18 T + 968 T^{2} + 15480 T^{3} + 516891 T^{4} + 15480 p^{2} T^{5} + 968 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \)
29$D_4\times C_2$ \( 1 - 2 T - 1571 T^{2} + 214 T^{3} + 1769980 T^{4} + 214 p^{2} T^{5} - 1571 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \)
31$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 18300 T^{3} + 1672334 T^{4} - 18300 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 + 68 T + 41 p T^{2} - 93168 T^{3} - 5925028 T^{4} - 93168 p^{2} T^{5} + 41 p^{5} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 7272 p^{2} T^{5} + 3461 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \)
43$C_2^2$ \( ( 1 + 90 T + 4549 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 68 T + 2312 T^{2} - 148716 T^{3} + 9565454 T^{4} - 148716 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 64 T + 4455 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 164 T + 6980 T^{2} + 551844 T^{3} - 68910913 T^{4} + 551844 p^{2} T^{5} + 6980 p^{4} T^{6} - 164 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 312604 p^{2} T^{5} + 5413 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 118 T + 10706 T^{2} + 763488 T^{3} + 51177839 T^{4} + 763488 p^{2} T^{5} + 10706 p^{4} T^{6} + 118 p^{6} T^{7} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 86 T + 4658 T^{2} - 258336 T^{3} + 1457087 T^{4} - 258336 p^{2} T^{5} + 4658 p^{4} T^{6} - 86 p^{6} T^{7} + p^{8} T^{8} \)
73$D_4\times C_2$ \( 1 - 58 T + 1682 T^{2} - 201144 T^{3} + 20590727 T^{4} - 201144 p^{2} T^{5} + 1682 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 - 20 T + 7290 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 188 T + 17672 T^{2} - 1935084 T^{3} + 200304482 T^{4} - 1935084 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 110 T + 12050 T^{2} + 1194180 T^{3} + 87746159 T^{4} + 1194180 p^{2} T^{5} + 12050 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 178 T + 13250 T^{2} - 318828 T^{3} - 39375793 T^{4} - 318828 p^{2} T^{5} + 13250 p^{4} T^{6} - 178 p^{6} T^{7} + p^{8} T^{8} \)
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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

−8.897265963844280738051635195991, −8.542406674399416479863623913156, −8.187717970619084706240089970434, −8.025267590201325047852963353808, −7.64581336876289724913851810123, −7.37392535168167102156522965516, −7.35293877964799800303952536886, −7.04781021017144469147386025599, −6.64485818598042187451591156882, −6.52519183433215440268242035846, −6.47042385538700673748647163492, −5.77162594505226814111926294650, −5.22443965129022885370420209795, −4.99293138979589701450096187871, −4.70457820148883697087967659873, −4.16612778173220414541138519632, −4.12654413689996763251641616921, −3.92704309085966328923534570142, −3.32827530042157390633862005554, −3.22036782315527311023042290875, −2.90837842008013024283547090375, −2.19439110478646587387858484394, −2.03148460179365990235758435867, −0.60282096552459194094195201766, −0.41060316139942039118486261665, 0.41060316139942039118486261665, 0.60282096552459194094195201766, 2.03148460179365990235758435867, 2.19439110478646587387858484394, 2.90837842008013024283547090375, 3.22036782315527311023042290875, 3.32827530042157390633862005554, 3.92704309085966328923534570142, 4.12654413689996763251641616921, 4.16612778173220414541138519632, 4.70457820148883697087967659873, 4.99293138979589701450096187871, 5.22443965129022885370420209795, 5.77162594505226814111926294650, 6.47042385538700673748647163492, 6.52519183433215440268242035846, 6.64485818598042187451591156882, 7.04781021017144469147386025599, 7.35293877964799800303952536886, 7.37392535168167102156522965516, 7.64581336876289724913851810123, 8.025267590201325047852963353808, 8.187717970619084706240089970434, 8.542406674399416479863623913156, 8.897265963844280738051635195991

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