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Properties

Label 8-605e4-1.1-c1e4-0-10
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 2·3-s + 4·4-s + 4·5-s + 6·6-s − 11·7-s − 4·8-s − 4·9-s − 12·10-s − 8·12-s − 7·13-s + 33·14-s − 8·15-s + 2·16-s − 3·17-s + 12·18-s − 12·19-s + 16·20-s + 22·21-s − 9·23-s + 8·24-s + 10·25-s + 21·26-s + 11·27-s − 44·28-s + 8·29-s + 24·30-s + ⋯
L(s)  = 1  − 2.12·2-s − 1.15·3-s + 2·4-s + 1.78·5-s + 2.44·6-s − 4.15·7-s − 1.41·8-s − 4/3·9-s − 3.79·10-s − 2.30·12-s − 1.94·13-s + 8.81·14-s − 2.06·15-s + 1/2·16-s − 0.727·17-s + 2.82·18-s − 2.75·19-s + 3.57·20-s + 4.80·21-s − 1.87·23-s + 1.63·24-s + 2·25-s + 4.11·26-s + 2.11·27-s − 8.31·28-s + 1.48·29-s + 4.38·30-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 + 3 T + 5 T^{2} + 7 T^{3} + 11 T^{4} + 7 p T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 8 T^{2} + 13 T^{3} + 35 T^{4} + 13 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 67 T^{2} + 276 T^{3} + 845 T^{4} + 276 p T^{5} + 67 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 59 T^{2} + 264 T^{3} + 1185 T^{4} + 264 p T^{5} + 59 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 60 T^{2} + 127 T^{3} + 1451 T^{4} + 127 p T^{5} + 60 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 122 T^{2} + 749 T^{3} + 3939 T^{4} + 749 p T^{5} + 122 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 38 T^{2} - 85 T^{3} - 979 T^{4} - 85 p T^{5} + 38 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 112 T^{2} - 601 T^{3} + 4759 T^{4} - 601 p T^{5} + 112 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 3 p T^{2} - 276 T^{3} + 3945 T^{4} - 276 p T^{5} + 3 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 92 T^{2} + 305 T^{3} + 4221 T^{4} + 305 p T^{5} + 92 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 118 T^{2} - 479 T^{3} + 5815 T^{4} - 479 p T^{5} + 118 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 293 T^{2} + 2900 T^{3} + 21441 T^{4} + 2900 p T^{5} + 293 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 137 T^{2} + 290 T^{3} + 8531 T^{4} + 290 p T^{5} + 137 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 245 T^{2} + 1776 T^{3} + 20351 T^{4} + 1776 p T^{5} + 245 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 127 T^{2} + 44 T^{3} + 4999 T^{4} + 44 p T^{5} + 127 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 203 T^{2} + 642 T^{3} + 17379 T^{4} + 642 p T^{5} + 203 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + T + 186 T^{2} - 37 T^{3} + 15845 T^{4} - 37 p T^{5} + 186 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 186 T^{2} + 925 T^{3} + 8531 T^{4} + 925 p T^{5} + 186 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 218 T^{2} - 1375 T^{3} + 20781 T^{4} - 1375 p T^{5} + 218 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 220 T^{2} + 487 T^{3} + 20123 T^{4} + 487 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 375 T^{2} + 3710 T^{3} + 48443 T^{4} + 3710 p T^{5} + 375 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 206 T^{2} - 400 T^{3} + 21551 T^{4} - 400 p T^{5} + 206 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 332 T^{2} - 1836 T^{3} + 45565 T^{4} - 1836 p T^{5} + 332 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−8.462047378476430151306586448767, −7.957323475028413984621733802382, −7.80150113134275490024170060266, −7.46628786548353133292239300866, −6.93343122810109517167153674457, −6.71650358263646396965133400056, −6.67418225543118668489603830253, −6.47852259253136746386084737987, −6.44158011250738377951481792855, −6.06207376195984481242308647853, −5.99098064131216675876197010619, −5.82068370482139506595601592947, −5.59070383351561562682853477032, −5.00230335305934554827261152232, −4.78372736270906072334256046136, −4.59721434802858480962714906620, −4.28076921666565554647974560291, −3.60177501393623652959348241926, −3.45976653402416795674018223528, −2.96010323474446574344077691806, −2.80134122159274896533293281679, −2.58738402693973646110576449948, −2.33687375240713757191578207116, −1.94863680280526318005072684132, −1.33977056843985594276932931220, 0, 0, 0, 0, 1.33977056843985594276932931220, 1.94863680280526318005072684132, 2.33687375240713757191578207116, 2.58738402693973646110576449948, 2.80134122159274896533293281679, 2.96010323474446574344077691806, 3.45976653402416795674018223528, 3.60177501393623652959348241926, 4.28076921666565554647974560291, 4.59721434802858480962714906620, 4.78372736270906072334256046136, 5.00230335305934554827261152232, 5.59070383351561562682853477032, 5.82068370482139506595601592947, 5.99098064131216675876197010619, 6.06207376195984481242308647853, 6.44158011250738377951481792855, 6.47852259253136746386084737987, 6.67418225543118668489603830253, 6.71650358263646396965133400056, 6.93343122810109517167153674457, 7.46628786548353133292239300866, 7.80150113134275490024170060266, 7.957323475028413984621733802382, 8.462047378476430151306586448767

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