Properties

Label 8-384e4-1.1-c1e4-0-0
Degree $8$
Conductor $21743271936$
Sign $1$
Analytic cond. $88.3961$
Root an. cond. $1.75107$
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 − 32·23-s − 4·25-s + 12·49-s − 32·71-s − 40·73-s − 5·81-s − 24·97-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s − 64·207-s + ⋯
L(s)  = 1  + 2/3·9-s − 6.67·23-s − 4/5·25-s + 12/7·49-s − 3.79·71-s − 4.68·73-s − 5/9·81-s − 2.43·97-s + 3.27·121-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 + 1.53·169-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 + 0.0708·199-s − 4.44·207-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{28} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(88.3961\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{28} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.5343293792\)
\(L(\frac12)\) \(\approx\) \(0.5343293792\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
good5$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
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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

−8.265909017699831400638512889083, −7.947766604517057122997277895903, −7.58377505561171798852967983053, −7.51671614009728113911271684319, −7.42819174845316496691319549424, −7.13901766699093494263261841770, −6.51420947557279667385486354534, −6.49874924579374396926734622456, −5.99830149779478165528715613354, −5.97878864229280295301841996051, −5.75888505131490339091231454492, −5.55266856522322212854624847045, −5.30570611399189772799361049528, −4.48105143546737670939677393280, −4.31013749661651041595736814992, −4.25126178719914202565305768732, −4.09543316838642553122278100661, −3.86827606007007513778805288423, −3.17912207350930069820924720373, −3.00006713872455957799728131876, −2.49080542961897640025428532955, −1.97827223360558049360806343969, −1.73879449695613555507946971694, −1.59836086660238128205971314312, −0.26711616371200848239957197239, 0.26711616371200848239957197239, 1.59836086660238128205971314312, 1.73879449695613555507946971694, 1.97827223360558049360806343969, 2.49080542961897640025428532955, 3.00006713872455957799728131876, 3.17912207350930069820924720373, 3.86827606007007513778805288423, 4.09543316838642553122278100661, 4.25126178719914202565305768732, 4.31013749661651041595736814992, 4.48105143546737670939677393280, 5.30570611399189772799361049528, 5.55266856522322212854624847045, 5.75888505131490339091231454492, 5.97878864229280295301841996051, 5.99830149779478165528715613354, 6.49874924579374396926734622456, 6.51420947557279667385486354534, 7.13901766699093494263261841770, 7.42819174845316496691319549424, 7.51671614009728113911271684319, 7.58377505561171798852967983053, 7.947766604517057122997277895903, 8.265909017699831400638512889083

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