Properties

Label 8-2352e4-1.1-c1e4-0-6
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $124410.$
Root an. cond. $4.33368$
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  + 5·9-s + 16·13-s − 2·25-s − 4·37-s − 12·61-s − 28·73-s + 16·81-s − 32·97-s + 20·109-s + 80·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 108·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 5/3·9-s + 4.43·13-s − 2/5·25-s − 0.657·37-s − 1.53·61-s − 3.27·73-s + 16/9·81-s − 3.24·97-s + 1.91·109-s + 7.39·117-s − 2·121-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 + 8.30·169-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(124410.\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.018619873\)
\(L(\frac12)\) \(\approx\) \(3.018619873\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2}( 1 + 3 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 23 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 35 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 95 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 107 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 167 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
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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

−6.38138251939944851980219349958, −6.11538298672732416983799848957, −5.84259545461250534936219790857, −5.77399626508176851761961432814, −5.76940372992010244081191866557, −5.62983841764269168220642875609, −4.95205519805719681484466041083, −4.77913634018938331496041938545, −4.68696232001401590122359276730, −4.57934455026507353882062436082, −4.07675278987242159295344336847, −3.97147758249226579107430812738, −3.80365606417093725535866851125, −3.62892121347177518730254026326, −3.48877681944885302994491703911, −3.17337151091118136168354972091, −2.96126512595993418245161730918, −2.46671361603398062678100262791, −2.37320654341794681629595739375, −1.83974472691206594939570096136, −1.55834919507136475650581347254, −1.22256879844287843679056171104, −1.20762295331120068890166440811, −1.19423749366218509530302355984, −0.24510044715582581986988388617, 0.24510044715582581986988388617, 1.19423749366218509530302355984, 1.20762295331120068890166440811, 1.22256879844287843679056171104, 1.55834919507136475650581347254, 1.83974472691206594939570096136, 2.37320654341794681629595739375, 2.46671361603398062678100262791, 2.96126512595993418245161730918, 3.17337151091118136168354972091, 3.48877681944885302994491703911, 3.62892121347177518730254026326, 3.80365606417093725535866851125, 3.97147758249226579107430812738, 4.07675278987242159295344336847, 4.57934455026507353882062436082, 4.68696232001401590122359276730, 4.77913634018938331496041938545, 4.95205519805719681484466041083, 5.62983841764269168220642875609, 5.76940372992010244081191866557, 5.77399626508176851761961432814, 5.84259545461250534936219790857, 6.11538298672732416983799848957, 6.38138251939944851980219349958

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