Properties

Label 8-1232e4-1.1-c1e4-0-6
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $9365.93$
Root an. cond. $3.13649$
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  + 2·9-s + 4·11-s + 20·23-s + 10·25-s + 44·37-s + 4·49-s + 32·53-s − 36·67-s + 4·71-s − 15·81-s + 8·99-s − 12·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 36·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2/3·9-s + 1.20·11-s + 4.17·23-s + 2·25-s + 7.23·37-s + 4/7·49-s + 4.39·53-s − 4.39·67-s + 0.474·71-s − 5/3·81-s + 0.804·99-s − 1.12·113-s − 0.909·121-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 + 0.0783·163-s + 0.0773·167-s − 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(9365.93\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.047542783\)
\(L(\frac12)\) \(\approx\) \(8.047542783\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good3$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \)
5$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
59$C_2^2$ \( ( 1 - 113 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 + 9 T + p T^{2} )^{4} \)
71$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 164 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 133 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 51 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.97404152778727236804150751455, −6.81371536135315739384037676534, −6.56906006515629390506108517936, −6.13746400498338839629780572700, −6.13120832594302514319763091226, −6.05136116376138711208767382913, −5.51609745643045463276098271538, −5.38912660225615699199797908508, −5.22984579948193774283988694324, −4.80593677984837704029640291901, −4.51792045636363850794880734999, −4.49332448858261255057141816923, −4.18803180466652277580907468467, −4.11724166103676985020402474123, −3.88124427005840656420863255054, −3.20808473773921154589710632675, −2.97133004468516564503849921027, −2.96794231552647954781870008755, −2.73902885068759476310571843518, −2.34946366657262754533259118749, −2.13235024101576415339644440944, −1.16893147848130655877465831524, −1.13735646189470533455780803352, −1.08595405206762833264161296480, −0.76226848196857346744060608430, 0.76226848196857346744060608430, 1.08595405206762833264161296480, 1.13735646189470533455780803352, 1.16893147848130655877465831524, 2.13235024101576415339644440944, 2.34946366657262754533259118749, 2.73902885068759476310571843518, 2.96794231552647954781870008755, 2.97133004468516564503849921027, 3.20808473773921154589710632675, 3.88124427005840656420863255054, 4.11724166103676985020402474123, 4.18803180466652277580907468467, 4.49332448858261255057141816923, 4.51792045636363850794880734999, 4.80593677984837704029640291901, 5.22984579948193774283988694324, 5.38912660225615699199797908508, 5.51609745643045463276098271538, 6.05136116376138711208767382913, 6.13120832594302514319763091226, 6.13746400498338839629780572700, 6.56906006515629390506108517936, 6.81371536135315739384037676534, 6.97404152778727236804150751455

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