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Properties

Label 8-3006e4-1.1-c1e4-0-0
Degree $8$
Conductor $8.165\times 10^{13}$
Sign $1$
Analytic cond. $331943.$
Root an. cond. $4.89928$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 10·4-s − 5·5-s + 7-s − 20·8-s + 20·10-s + 8·13-s − 4·14-s + 35·16-s − 10·17-s − 2·19-s − 50·20-s − 2·23-s + 5·25-s − 32·26-s + 10·28-s − 14·29-s + 31-s − 56·32-s + 40·34-s − 5·35-s + 5·37-s + 8·38-s + 100·40-s − 14·41-s + 2·43-s + 8·46-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 2.23·5-s + 0.377·7-s − 7.07·8-s + 6.32·10-s + 2.21·13-s − 1.06·14-s + 35/4·16-s − 2.42·17-s − 0.458·19-s − 11.1·20-s − 0.417·23-s + 25-s − 6.27·26-s + 1.88·28-s − 2.59·29-s + 0.179·31-s − 9.89·32-s + 6.85·34-s − 0.845·35-s + 0.821·37-s + 1.29·38-s + 15.8·40-s − 2.18·41-s + 0.304·43-s + 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 167^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 167^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 167^{4}\)
Sign: $1$
Analytic conductor: \(331943.\)
Root analytic conductor: \(4.89928\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 167^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 + T )^{4} \)
3 \( 1 \)
167$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr S_4$ \( 1 + p T + 4 p T^{2} + 11 p T^{3} + 142 T^{4} + 11 p^{2} T^{5} + 4 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - T + 8 T^{2} - 3 p T^{3} + 78 T^{4} - 3 p^{2} T^{5} + 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2 \wr S_4$ \( 1 - 8 T + 56 T^{2} - 256 T^{3} + 1102 T^{4} - 256 p T^{5} + 56 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 10 T + 88 T^{2} + 494 T^{3} + 2398 T^{4} + 494 p T^{5} + 88 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 2 T + 44 T^{2} + 82 T^{3} + 1078 T^{4} + 82 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 2 T + 60 T^{2} + 106 T^{3} + 1830 T^{4} + 106 p T^{5} + 60 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 14 T + 152 T^{2} + 1162 T^{3} + 7102 T^{4} + 1162 p T^{5} + 152 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - T + 88 T^{2} + 3 T^{3} + 3470 T^{4} + 3 p T^{5} + 88 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 5 T + 82 T^{2} - 371 T^{3} + 3898 T^{4} - 371 p T^{5} + 82 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 14 T + 152 T^{2} + 1242 T^{3} + 8030 T^{4} + 1242 p T^{5} + 152 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 2 T + 140 T^{2} - 298 T^{3} + 8374 T^{4} - 298 p T^{5} + 140 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 7 T + 156 T^{2} + 779 T^{3} + 10182 T^{4} + 779 p T^{5} + 156 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 3 T + 172 T^{2} + 441 T^{3} + 12622 T^{4} + 441 p T^{5} + 172 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 5 T + 120 T^{2} - 921 T^{3} + 8006 T^{4} - 921 p T^{5} + 120 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 2 T + 208 T^{2} - 294 T^{3} + 18062 T^{4} - 294 p T^{5} + 208 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 7 T + 82 T^{2} - 441 T^{3} - 1198 T^{4} - 441 p T^{5} + 82 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 2 T + 116 T^{2} + 298 T^{3} + 13174 T^{4} + 298 p T^{5} + 116 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2 T + 160 T^{2} - 366 T^{3} + 13214 T^{4} - 366 p T^{5} + 160 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 10 T + 260 T^{2} + 1994 T^{3} + 30070 T^{4} + 1994 p T^{5} + 260 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 13 T + 236 T^{2} + 2465 T^{3} + 29470 T^{4} + 2465 p T^{5} + 236 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 13 T + 354 T^{2} + 2955 T^{3} + 45722 T^{4} + 2955 p T^{5} + 354 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 5 T + 242 T^{2} - 827 T^{3} + 31386 T^{4} - 827 p T^{5} + 242 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.74032547757760489969429421109, −6.53683481708450856963098024370, −6.26453561019014554338363971483, −6.15093957558764154903876995077, −6.08131107382062768736264667780, −5.73823406597983189949693834446, −5.35065253997113462685789840248, −5.27656393659833719421544388174, −5.13469981889554307292158489303, −4.47082089957616055992996377824, −4.45504665392988890756299492150, −4.31305290953678810042882856556, −4.01118482076769723664379969829, −3.68757852859161312793919658963, −3.60326075697916086481872890048, −3.48191675091042031613224596935, −3.36746979573927515562215205578, −2.68099815052239903953087550753, −2.52681086648930075552628745918, −2.45027557999848578464825754047, −2.06384254901616198896140584986, −1.68683543102571850675880246447, −1.45849595295937742726695224659, −1.27541584548142495746554201830, −1.11466830082390253149866671954, 0, 0, 0, 0, 1.11466830082390253149866671954, 1.27541584548142495746554201830, 1.45849595295937742726695224659, 1.68683543102571850675880246447, 2.06384254901616198896140584986, 2.45027557999848578464825754047, 2.52681086648930075552628745918, 2.68099815052239903953087550753, 3.36746979573927515562215205578, 3.48191675091042031613224596935, 3.60326075697916086481872890048, 3.68757852859161312793919658963, 4.01118482076769723664379969829, 4.31305290953678810042882856556, 4.45504665392988890756299492150, 4.47082089957616055992996377824, 5.13469981889554307292158489303, 5.27656393659833719421544388174, 5.35065253997113462685789840248, 5.73823406597983189949693834446, 6.08131107382062768736264667780, 6.15093957558764154903876995077, 6.26453561019014554338363971483, 6.53683481708450856963098024370, 6.74032547757760489969429421109

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