Properties

Degree $8$
Conductor $7.834\times 10^{15}$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{9408} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.872939413\)
\(L(\frac12)\) \(\approx\) \(8.872939413\)
\(L(\frac{3}{2})\) 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

−5.51862146007043986597832645094, −4.89743906608515189752734151749, −4.77839397162779953677479043158, −4.74422364175357524021677213000, −4.68324331258651842466753528589, −4.39759960044208314505424908862, −4.15641205810341767042592173261, −4.07968623477262768367888796387, −4.02413105192893999837081609166, −3.54812312964169985613364389130, −3.48456388054080826628237662221, −3.29217750645491835116996973254, −3.24599428325379023712870147718, −3.14679361859741723150044891959, −2.67954144253839376605831423293, −2.49567764639341584676263628731, −2.40935570778634565480645735524, −2.08884954134373650945003983125, −2.08108092775604292701173351646, −1.59058360785738361294948635896, −1.54579749207019016309165552653, −1.03955488540382184635414131454, −0.77223895971596238005368280968, −0.71670789242166631389331725933, −0.26470681966396797231641843767, 0.26470681966396797231641843767, 0.71670789242166631389331725933, 0.77223895971596238005368280968, 1.03955488540382184635414131454, 1.54579749207019016309165552653, 1.59058360785738361294948635896, 2.08108092775604292701173351646, 2.08884954134373650945003983125, 2.40935570778634565480645735524, 2.49567764639341584676263628731, 2.67954144253839376605831423293, 3.14679361859741723150044891959, 3.24599428325379023712870147718, 3.29217750645491835116996973254, 3.48456388054080826628237662221, 3.54812312964169985613364389130, 4.02413105192893999837081609166, 4.07968623477262768367888796387, 4.15641205810341767042592173261, 4.39759960044208314505424908862, 4.68324331258651842466753528589, 4.74422364175357524021677213000, 4.77839397162779953677479043158, 4.89743906608515189752734151749, 5.51862146007043986597832645094

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