Properties

Degree $8$
Conductor $1.736\times 10^{14}$
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  + 4·2-s + 4·3-s + 10·4-s − 4·5-s + 16·6-s + 7-s + 20·8-s + 10·9-s − 16·10-s + 40·12-s + 2·13-s + 4·14-s − 16·15-s + 35·16-s + 11·17-s + 40·18-s + 19-s − 40·20-s + 4·21-s + 80·24-s + 10·25-s + 8·26-s + 20·27-s + 10·28-s + 9·29-s − 64·30-s + 17·31-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 0.377·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s + 11.5·12-s + 0.554·13-s + 1.06·14-s − 4.13·15-s + 35/4·16-s + 2.66·17-s + 9.42·18-s + 0.229·19-s − 8.94·20-s + 0.872·21-s + 16.3·24-s + 2·25-s + 1.56·26-s + 3.84·27-s + 1.88·28-s + 1.67·29-s − 11.6·30-s + 3.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{3630} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(172.6462840\)
\(L(\frac12)\) \(\approx\) \(172.6462840\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good7$(((C_4 \times C_2): C_2):C_2):C_2$ \( 1 - T - T^{2} + 13 T^{3} + 4 T^{4} + 13 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 11 T^{2} - 36 T^{3} + 69 T^{4} - 36 p T^{5} + 11 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 89 T^{2} - 517 T^{3} + 2464 T^{4} - 517 p T^{5} + 89 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - T + 37 T^{2} - 3 T^{3} + 680 T^{4} - 3 p T^{5} + 37 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 27 T^{2} + 130 T^{3} + 239 T^{4} + 130 p T^{5} + 27 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 67 T^{2} - 297 T^{3} + 1480 T^{4} - 297 p T^{5} + 67 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 17 T + 203 T^{2} - 1619 T^{3} + 10500 T^{4} - 1619 p T^{5} + 203 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 77 T^{2} - 540 T^{3} + 4701 T^{4} - 540 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 35 T^{2} - 201 T^{3} - 56 p T^{4} - 201 p T^{5} + 35 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - T + 133 T^{2} - 75 T^{3} + 7736 T^{4} - 75 p T^{5} + 133 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 173 T^{2} - 1050 T^{3} + 11099 T^{4} - 1050 p T^{5} + 173 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 213 T^{2} - 1685 T^{3} + 16916 T^{4} - 1685 p T^{5} + 213 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 144 T^{2} + 1670 T^{3} + 11406 T^{4} + 1670 p T^{5} + 144 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 179 T^{2} - 1883 T^{3} + 15084 T^{4} - 1883 p T^{5} + 179 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 224 T^{2} - 1650 T^{3} + 23006 T^{4} - 1650 p T^{5} + 224 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$D_{4}$ \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 115 T^{2} + 727 T^{3} + 14764 T^{4} + 727 p T^{5} + 115 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 23 T + 435 T^{2} + 4963 T^{3} + 54864 T^{4} + 4963 p T^{5} + 435 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 252 T^{2} + 910 T^{3} + 31206 T^{4} + 910 p T^{5} + 252 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.08454709983431390268044812026, −5.60933914048011188207245608817, −5.43634951923656523618602256567, −5.38695492397441688142152045062, −5.29231075629187107178354167482, −4.77171184154178775765374398901, −4.66448898325768985640712959115, −4.54663916375618092336784721960, −4.48077183998019012245543089234, −3.98456098404068093739145477486, −3.87216660369152442103566152108, −3.86562410746253465151529381445, −3.78064925181591672341798027975, −3.24140481015611356322499984565, −3.17612640118119707047025608070, −3.04251636807936627189316320927, −2.90196480188723086541213408767, −2.64646573201126995977151711303, −2.44126164499413824530116810989, −2.07836983073051770458683916511, −1.99955995047539241488495197383, −1.26729446496382611540885515928, −1.20105954923881766848373187949, −0.932887620285584336683531344424, −0.815355966158160828379332981311, 0.815355966158160828379332981311, 0.932887620285584336683531344424, 1.20105954923881766848373187949, 1.26729446496382611540885515928, 1.99955995047539241488495197383, 2.07836983073051770458683916511, 2.44126164499413824530116810989, 2.64646573201126995977151711303, 2.90196480188723086541213408767, 3.04251636807936627189316320927, 3.17612640118119707047025608070, 3.24140481015611356322499984565, 3.78064925181591672341798027975, 3.86562410746253465151529381445, 3.87216660369152442103566152108, 3.98456098404068093739145477486, 4.48077183998019012245543089234, 4.54663916375618092336784721960, 4.66448898325768985640712959115, 4.77171184154178775765374398901, 5.29231075629187107178354167482, 5.38695492397441688142152045062, 5.43634951923656523618602256567, 5.60933914048011188207245608817, 6.08454709983431390268044812026

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