Properties

Label 8-1216e4-1.1-c1e4-0-8
Degree $8$
Conductor $2.186\times 10^{12}$
Sign $1$
Analytic cond. $8888.79$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·9-s − 12·17-s + 14·25-s + 24·41-s − 22·49-s − 4·73-s − 6·81-s + 16·97-s − 72·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 4/3·9-s − 2.91·17-s + 14/5·25-s + 3.74·41-s − 3.14·49-s − 0.468·73-s − 2/3·81-s + 1.62·97-s − 6.77·113-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(8888.79\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1216} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.749143263\)
\(L(\frac12)\) \(\approx\) \(2.749143263\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 130 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
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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

−6.90739691683867808223964668136, −6.67083769185318224221616767988, −6.54061604236891968259855100544, −6.50430011219516939125375699798, −6.20004686601810393127131810002, −5.97740398979658914658892958820, −5.52567955327388282498926208982, −5.30996495404645124766230089083, −5.13484712119125654347457659588, −4.91954337306519218008702268791, −4.57629126388276573532465852878, −4.36641153913946972863328227426, −4.26152648992979132034159970546, −4.04664179287802178979533292477, −4.01092049735730250287419843736, −3.30172783207716918971575777861, −3.05900881158751192129317369803, −2.79525138281833089959285908556, −2.78433399964219420287668859552, −2.14464037007539385288493675998, −2.10934568023208294799193817828, −1.58264610748522764387088759791, −1.34366561065017568858271144626, −0.869532506126091502286124386544, −0.38882321693785586860652856629, 0.38882321693785586860652856629, 0.869532506126091502286124386544, 1.34366561065017568858271144626, 1.58264610748522764387088759791, 2.10934568023208294799193817828, 2.14464037007539385288493675998, 2.78433399964219420287668859552, 2.79525138281833089959285908556, 3.05900881158751192129317369803, 3.30172783207716918971575777861, 4.01092049735730250287419843736, 4.04664179287802178979533292477, 4.26152648992979132034159970546, 4.36641153913946972863328227426, 4.57629126388276573532465852878, 4.91954337306519218008702268791, 5.13484712119125654347457659588, 5.30996495404645124766230089083, 5.52567955327388282498926208982, 5.97740398979658914658892958820, 6.20004686601810393127131810002, 6.50430011219516939125375699798, 6.54061604236891968259855100544, 6.67083769185318224221616767988, 6.90739691683867808223964668136

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