Properties

Label 8-4704e4-1.1-c1e4-0-4
Degree $8$
Conductor $4.896\times 10^{14}$
Sign $1$
Analytic cond. $1.99057\times 10^{6}$
Root an. cond. $6.12875$
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  + 4·3-s − 4·5-s + 10·9-s − 4·11-s − 8·13-s − 16·15-s − 4·17-s + 8·19-s − 12·23-s + 4·25-s + 20·27-s − 8·31-s − 16·33-s − 32·39-s − 20·41-s + 8·43-s − 40·45-s − 16·47-s − 16·51-s + 16·55-s + 32·57-s − 16·61-s + 32·65-s + 8·67-s − 48·69-s − 12·71-s − 8·73-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.78·5-s + 10/3·9-s − 1.20·11-s − 2.21·13-s − 4.13·15-s − 0.970·17-s + 1.83·19-s − 2.50·23-s + 4/5·25-s + 3.84·27-s − 1.43·31-s − 2.78·33-s − 5.12·39-s − 3.12·41-s + 1.21·43-s − 5.96·45-s − 2.33·47-s − 2.24·51-s + 2.15·55-s + 4.23·57-s − 2.04·61-s + 3.96·65-s + 0.977·67-s − 5.77·69-s − 1.42·71-s − 0.936·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.99057\times 10^{6}\)
Root analytic conductor: \(6.12875\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4704} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 12 T^{2} + 36 T^{3} + 98 T^{4} + 36 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 24 T^{2} + 84 T^{3} + 302 T^{4} + 84 p T^{5} + 24 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 48 T^{2} + 232 T^{3} + 978 T^{4} + 232 p T^{5} + 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 44 T^{2} + 84 T^{3} + 818 T^{4} + 84 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 68 T^{2} - 392 T^{3} + 1926 T^{4} - 392 p T^{5} + 68 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 120 T^{2} + 780 T^{3} + 4350 T^{4} + 780 p T^{5} + 120 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 60 T^{2} + 128 T^{3} + 1862 T^{4} + 128 p T^{5} + 60 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 84 T^{2} + 616 T^{3} + 3222 T^{4} + 616 p T^{5} + 84 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 220 T^{2} + 1540 T^{3} + 9874 T^{4} + 1540 p T^{5} + 220 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} - 648 T^{3} + 6710 T^{4} - 648 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 164 T^{2} + 1232 T^{3} + 170 p T^{4} + 1232 p T^{5} + 164 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 140 T^{2} - 128 T^{3} + 9494 T^{4} - 128 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 84 T^{2} + 960 T^{3} + 1350 T^{4} + 960 p T^{5} + 84 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 288 T^{2} + 2768 T^{3} + 27282 T^{4} + 2768 p T^{5} + 288 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 124 T^{2} + 56 T^{3} + 3286 T^{4} + 56 p T^{5} + 124 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 248 T^{2} + 2380 T^{3} + 25406 T^{4} + 2380 p T^{5} + 248 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 256 T^{2} + 1608 T^{3} + 27170 T^{4} + 1608 p T^{5} + 256 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 284 T^{2} + 2768 T^{3} + 31366 T^{4} + 2768 p T^{5} + 284 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 204 T^{2} + 256 T^{3} + 21878 T^{4} + 256 p T^{5} + 204 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 524 T^{2} + 76 p T^{3} + 72658 T^{4} + 76 p^{2} T^{5} + 524 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 40 T + 896 T^{2} + 13608 T^{3} + 153858 T^{4} + 13608 p T^{5} + 896 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.53178432971048629942733907083, −5.79228417998789422882098580441, −5.65992689347789687994185762287, −5.64297393998205658208719770816, −5.60523225643800821280770624437, −5.10251910583888918691392685393, −5.05316975212240775099868541809, −4.73298237209800890557691678190, −4.63952369988187650605023347117, −4.17307058392455019038916741738, −4.14413083117494025497564057097, −4.13169057056592715118058325792, −4.08904586025415098216020353067, −3.44367213109694849847679642344, −3.34156022818580945903766204010, −3.24058377505302159051935385357, −3.12590709057951200610866405755, −2.74008847810913372188542706899, −2.56443177363923014464589155469, −2.48920717657099202327817235417, −2.17963354050940444829501882332, −1.83936953167467577440789140491, −1.60779621625546602376837376713, −1.36612095705292655408578235408, −1.25777636302300575573719250321, 0, 0, 0, 0, 1.25777636302300575573719250321, 1.36612095705292655408578235408, 1.60779621625546602376837376713, 1.83936953167467577440789140491, 2.17963354050940444829501882332, 2.48920717657099202327817235417, 2.56443177363923014464589155469, 2.74008847810913372188542706899, 3.12590709057951200610866405755, 3.24058377505302159051935385357, 3.34156022818580945903766204010, 3.44367213109694849847679642344, 4.08904586025415098216020353067, 4.13169057056592715118058325792, 4.14413083117494025497564057097, 4.17307058392455019038916741738, 4.63952369988187650605023347117, 4.73298237209800890557691678190, 5.05316975212240775099868541809, 5.10251910583888918691392685393, 5.60523225643800821280770624437, 5.64297393998205658208719770816, 5.65992689347789687994185762287, 5.79228417998789422882098580441, 6.53178432971048629942733907083

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