Properties

Label 8-240e4-1.1-c1e4-0-6
Degree $8$
Conductor $3317760000$
Sign $1$
Analytic cond. $13.4881$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·7-s + 6·9-s + 4·13-s − 16·15-s − 4·17-s − 16·21-s + 12·23-s + 2·25-s + 4·27-s + 8·29-s + 16·35-s − 12·37-s − 16·39-s − 4·43-s + 24·45-s + 20·47-s + 8·49-s + 16·51-s + 12·53-s − 16·59-s + 24·61-s + 24·63-s + 16·65-s − 12·67-s − 48·69-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s + 1.51·7-s + 2·9-s + 1.10·13-s − 4.13·15-s − 0.970·17-s − 3.49·21-s + 2.50·23-s + 2/5·25-s + 0.769·27-s + 1.48·29-s + 2.70·35-s − 1.97·37-s − 2.56·39-s − 0.609·43-s + 3.57·45-s + 2.91·47-s + 8/7·49-s + 2.24·51-s + 1.64·53-s − 2.08·59-s + 3.07·61-s + 3.02·63-s + 1.98·65-s − 1.46·67-s − 5.77·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(13.4881\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.371571912\)
\(L(\frac12)\) \(\approx\) \(1.371571912\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good7$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 20 T^{3} + 46 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$D_4\times C_2$ \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} + 4 T^{3} - 194 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 178 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$D_4\times C_2$ \( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
23$D_4\times C_2$ \( 1 - 12 T + 72 T^{2} - 444 T^{3} + 2542 T^{4} - 444 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} + 468 T^{3} + 3038 T^{4} + 468 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 164 T^{3} + 3358 T^{4} + 164 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 20 T + 200 T^{2} - 1860 T^{3} + 15182 T^{4} - 1860 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
61$D_{4}$ \( ( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 12 T + 72 T^{2} - 180 T^{3} - 6274 T^{4} - 180 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
71$D_4\times C_2$ \( 1 - 132 T^{2} + 13286 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \)
73$D_4\times C_2$ \( 1 - 4 T + 8 T^{2} - 44 T^{3} - 3602 T^{4} - 44 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 - 292 T^{2} + 33670 T^{4} - 292 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 + 4 T + 8 T^{2} + 196 T^{3} + 3646 T^{4} + 196 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.829071249201966031866485214267, −8.688698113463902045504923939393, −8.501945218545773747384785900997, −8.011343554578811119457739131543, −8.007464440794589597600333635888, −7.15492821411022272513245463474, −7.04321969015559544984800931797, −7.00409794465703421322718669228, −6.56522091801796313853129072653, −6.52130409254316779265758767795, −5.80758421365607720424360520109, −5.74728602771783253441411456528, −5.74440843936315810179120356496, −5.42126457635478722499873033198, −5.06975492303816037231165127328, −4.75237944573905987417453971246, −4.70765864556558544360177137702, −4.07976335334290891779966422438, −3.93474741741486724813910761269, −3.14442093606505853477105600409, −2.67274229363317024227521916753, −2.44633282563865214618409515269, −1.69741552337889595398284927044, −1.36317808131470439466068699911, −0.849697606789767347319049905355, 0.849697606789767347319049905355, 1.36317808131470439466068699911, 1.69741552337889595398284927044, 2.44633282563865214618409515269, 2.67274229363317024227521916753, 3.14442093606505853477105600409, 3.93474741741486724813910761269, 4.07976335334290891779966422438, 4.70765864556558544360177137702, 4.75237944573905987417453971246, 5.06975492303816037231165127328, 5.42126457635478722499873033198, 5.74440843936315810179120356496, 5.74728602771783253441411456528, 5.80758421365607720424360520109, 6.52130409254316779265758767795, 6.56522091801796313853129072653, 7.00409794465703421322718669228, 7.04321969015559544984800931797, 7.15492821411022272513245463474, 8.007464440794589597600333635888, 8.011343554578811119457739131543, 8.501945218545773747384785900997, 8.688698113463902045504923939393, 8.829071249201966031866485214267

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