Properties

Label 8-378e4-1.1-c2e4-0-5
Degree $8$
Conductor $20415837456$
Sign $1$
Analytic cond. $11254.0$
Root an. cond. $3.20932$
Motivic weight $2$
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  + 4·4-s − 28·7-s + 12·16-s + 46·25-s − 112·28-s + 100·37-s − 164·43-s + 490·49-s + 32·64-s + 248·67-s − 412·79-s + 184·100-s + 100·109-s − 336·112-s − 340·121-s + 127-s + 131-s + 137-s + 139-s + 400·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 244·169-s − 656·172-s + ⋯
L(s)  = 1  + 4-s − 4·7-s + 3/4·16-s + 1.83·25-s − 4·28-s + 2.70·37-s − 3.81·43-s + 10·49-s + 1/2·64-s + 3.70·67-s − 5.21·79-s + 1.83·100-s + 0.917·109-s − 3·112-s − 2.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 2.70·148-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.44·169-s − 3.81·172-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{12} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(11254.0\)
Root analytic conductor: \(3.20932\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{12} \cdot 7^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.444537847\)
\(L(\frac12)\) \(\approx\) \(1.444537847\)
\(L(2)\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good5$C_2^2$ \( ( 1 - 23 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 122 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 551 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 142 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 410 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 1706 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 25 T + p^{2} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 3335 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 41 T + p^{2} T^{2} )^{4} \)
47$C_2^2$ \( ( 1 + 3385 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5639 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 3142 T^{2} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 5474 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 5258 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 103 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 12455 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 17954 T^{2} + p^{4} T^{4} )^{2} \)
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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

−8.095325820832517843477295123427, −7.79549905758922416287813150209, −7.19604906094848634221927527121, −7.09763645821544701379276371791, −6.91149560996077720553074750637, −6.73691251300719521685977029397, −6.54145753819380707711820345334, −6.46361785393788388154587982822, −5.96994186368367125496755330699, −5.81864386708286438057418575372, −5.69292502807773075081909088973, −5.22161293670829254788645492047, −4.97509430552215828889927558783, −4.45809600061423475236924838234, −4.08886739506825267781690323318, −3.92279269697685193369062673752, −3.41292162345846834917181682422, −3.28412238873535732536608356710, −2.97708618840699619322199107373, −2.69047633780326687437804106914, −2.61839015641032715690427938738, −2.00720948699664314698553377449, −1.33951427476229900711892300759, −0.76231019221892289713949673209, −0.32571508825397390698992131762, 0.32571508825397390698992131762, 0.76231019221892289713949673209, 1.33951427476229900711892300759, 2.00720948699664314698553377449, 2.61839015641032715690427938738, 2.69047633780326687437804106914, 2.97708618840699619322199107373, 3.28412238873535732536608356710, 3.41292162345846834917181682422, 3.92279269697685193369062673752, 4.08886739506825267781690323318, 4.45809600061423475236924838234, 4.97509430552215828889927558783, 5.22161293670829254788645492047, 5.69292502807773075081909088973, 5.81864386708286438057418575372, 5.96994186368367125496755330699, 6.46361785393788388154587982822, 6.54145753819380707711820345334, 6.73691251300719521685977029397, 6.91149560996077720553074750637, 7.09763645821544701379276371791, 7.19604906094848634221927527121, 7.79549905758922416287813150209, 8.095325820832517843477295123427

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