Properties

Label 8-384e4-1.1-c5e4-0-2
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $1.43868\times 10^{7}$
Root an. cond. $7.84776$
Motivic weight $5$
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  + 482·9-s − 1.25e4·25-s + 6.72e4·49-s − 2.01e5·73-s + 1.73e5·81-s + 3.41e5·97-s + 1.94e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.48e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 6.02e6·225-s + ⋯
L(s)  = 1  + 1.98·9-s − 4·25-s + 4·49-s − 4.42·73-s + 2.93·81-s + 3.68·97-s + 1.20·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 4·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + 1.54e−6·211-s + 1.34e−6·223-s − 7.93·225-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1.43868\times 10^{7}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(5.733456880\)
\(L(\frac12)\) \(\approx\) \(5.733456880\)
\(L(\frac{7}{2})\) 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$C_2^2$ \( 1 - 482 T^{2} + p^{10} T^{4} \)
good5$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
7$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
11$C_2^2$ \( ( 1 - 97426 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
17$C_2$ \( ( 1 - 1914 T + p^{5} T^{2} )^{2}( 1 + 1914 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( ( 1 + 3353726 T^{2} + p^{10} T^{4} )^{2} \)
23$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
29$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
31$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
37$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
41$C_2$ \( ( 1 - 13926 T + p^{5} T^{2} )^{2}( 1 + 13926 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( ( 1 + 214485614 T^{2} + p^{10} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
53$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
59$C_2^2$ \( ( 1 + 921043598 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 1813708382 T^{2} + p^{10} T^{4} )^{2} \)
71$C_2$ \( ( 1 + p^{5} T^{2} )^{4} \)
73$C_2$ \( ( 1 + 50402 T + p^{5} T^{2} )^{4} \)
79$C_2$ \( ( 1 - p^{5} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 96051518 T^{2} + p^{10} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 7218 T + p^{5} T^{2} )^{2}( 1 + 7218 T + p^{5} T^{2} )^{2} \)
97$C_2$ \( ( 1 - 85450 T + p^{5} T^{2} )^{4} \)
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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

−7.41936783811344096607516252619, −7.17917941736858277104977146565, −7.10228691476581801793932003813, −6.61613809907756844995687367617, −6.18039074858922837520975255443, −6.13526461888642723655798441791, −5.84088195826473587930625622708, −5.69260834548156464518648493017, −5.36750540088211356053861752192, −4.99926058637496078519557836094, −4.59540894618847607564137499827, −4.43359940849658964561248220190, −4.08402979412235295218515030413, −4.00643561877099630690593931984, −3.78600850639022093635281639807, −3.32898853573931163213926122579, −3.12452867464138382037063581273, −2.41639972827019104445423530293, −2.30725099578983947053391483440, −2.02423555220664885187159177332, −1.55529597136221000885224437089, −1.50137340433495315001704176978, −0.956754037727464349758726201996, −0.49218236380650039149897255778, −0.35718485082229875548867360965, 0.35718485082229875548867360965, 0.49218236380650039149897255778, 0.956754037727464349758726201996, 1.50137340433495315001704176978, 1.55529597136221000885224437089, 2.02423555220664885187159177332, 2.30725099578983947053391483440, 2.41639972827019104445423530293, 3.12452867464138382037063581273, 3.32898853573931163213926122579, 3.78600850639022093635281639807, 4.00643561877099630690593931984, 4.08402979412235295218515030413, 4.43359940849658964561248220190, 4.59540894618847607564137499827, 4.99926058637496078519557836094, 5.36750540088211356053861752192, 5.69260834548156464518648493017, 5.84088195826473587930625622708, 6.13526461888642723655798441791, 6.18039074858922837520975255443, 6.61613809907756844995687367617, 7.10228691476581801793932003813, 7.17917941736858277104977146565, 7.41936783811344096607516252619

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