Properties

Label 8-320e4-1.1-c1e4-0-4
Degree $8$
Conductor $10485760000$
Sign $1$
Analytic cond. $42.6293$
Root an. cond. $1.59850$
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  + 12·7-s + 4·9-s + 12·23-s − 2·25-s − 24·31-s + 24·41-s − 12·47-s + 68·49-s + 48·63-s + 24·71-s − 16·73-s − 48·79-s + 6·81-s − 24·89-s + 16·97-s + 12·103-s − 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 144·161-s + 163-s + ⋯
L(s)  = 1  + 4.53·7-s + 4/3·9-s + 2.50·23-s − 2/5·25-s − 4.31·31-s + 3.74·41-s − 1.75·47-s + 68/7·49-s + 6.04·63-s + 2.84·71-s − 1.87·73-s − 5.40·79-s + 2/3·81-s − 2.54·89-s + 1.62·97-s + 1.18·103-s − 2.25·113-s + 1.81·121-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 + 0.0798·157-s + 11.3·161-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(42.6293\)
Root analytic conductor: \(1.59850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{320} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.607815876\)
\(L(\frac12)\) \(\approx\) \(4.607815876\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$D_{4}$ \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{4} \)
31$D_{4}$ \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 68 T^{2} + 2154 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \)
47$D_{4}$ \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 - 76 T^{2} + 1974 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 164 T^{2} + 13002 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 70 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
83$D_4\times C_2$ \( 1 - 308 T^{2} + 37386 T^{4} - 308 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.262800374752680871730734032005, −8.260522407781695279026341291730, −7.81485946296382063015285684623, −7.71261604603941745553429114516, −7.58721003950842959170604791400, −7.28482561540555240027552186332, −7.03653098665293352722038615522, −6.91245819544501300939434490426, −6.57404807877663760465410882540, −5.76482792120193023122931491328, −5.56678722007914872470340498459, −5.55839792909619361598525833106, −5.40921528857901648583958074354, −4.74923742536354138952265870981, −4.71955098095264012975912458241, −4.46653452313539574964867108407, −4.30954755324240951155302494286, −3.97851902355099256593992072877, −3.44363050086677859955276502146, −3.10347785620104204351078802474, −2.31690934466756140279194415739, −2.20992760395063336859024110249, −1.54913821355366333262453587199, −1.44953091408692283366044663361, −1.22447303076733250374926377475, 1.22447303076733250374926377475, 1.44953091408692283366044663361, 1.54913821355366333262453587199, 2.20992760395063336859024110249, 2.31690934466756140279194415739, 3.10347785620104204351078802474, 3.44363050086677859955276502146, 3.97851902355099256593992072877, 4.30954755324240951155302494286, 4.46653452313539574964867108407, 4.71955098095264012975912458241, 4.74923742536354138952265870981, 5.40921528857901648583958074354, 5.55839792909619361598525833106, 5.56678722007914872470340498459, 5.76482792120193023122931491328, 6.57404807877663760465410882540, 6.91245819544501300939434490426, 7.03653098665293352722038615522, 7.28482561540555240027552186332, 7.58721003950842959170604791400, 7.71261604603941745553429114516, 7.81485946296382063015285684623, 8.260522407781695279026341291730, 8.262800374752680871730734032005

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