Properties

Label 8-2352e4-1.1-c1e4-0-16
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  − 4·3-s + 6·9-s + 12·25-s + 4·27-s + 24·37-s + 24·47-s + 48·59-s − 48·75-s − 37·81-s + 24·83-s + 48·109-s − 96·111-s − 8·121-s + 127-s + 131-s + 137-s + 139-s − 96·141-s + 149-s + 151-s + 157-s + 163-s + 167-s − 48·169-s + 173-s − 192·177-s + 179-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s + 12/5·25-s + 0.769·27-s + 3.94·37-s + 3.50·47-s + 6.24·59-s − 5.54·75-s − 4.11·81-s + 2.63·83-s + 4.59·109-s − 9.11·111-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.08·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.69·169-s + 0.0760·173-s − 14.4·177-s + 0.0747·179-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

\(L(1)\) \(\approx\) \(2.641006331\)
\(L(\frac12)\) \(\approx\) \(2.641006331\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 + 120 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 48 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
89$C_2^2$ \( ( 1 - 174 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.40690404072027257896412183166, −5.98670371741266532482946664618, −5.93303045254508675153672203626, −5.84522497395381685060709167758, −5.64278476957821505690131161406, −5.27088430167978021327512409620, −5.19544374056313495832522896435, −4.94113669196557769937134841000, −4.89955661027194404829139866549, −4.61032214450430568109449426950, −4.24308495911181245228612306380, −4.13113444317770751652418728048, −3.98676849454886116192549718051, −3.56986651378450047748906429905, −3.51799200898076214393234575957, −2.96179013381296062116482403623, −2.69138847095621900030411632543, −2.52628119346454028785751727899, −2.39156947068251140142355804482, −2.19570800483541272806441404995, −1.54770447088838743360439621952, −0.947312897402252552708280492524, −0.886425458672409998414548296656, −0.811571882214165232446931840180, −0.50138539536290904658458147728, 0.50138539536290904658458147728, 0.811571882214165232446931840180, 0.886425458672409998414548296656, 0.947312897402252552708280492524, 1.54770447088838743360439621952, 2.19570800483541272806441404995, 2.39156947068251140142355804482, 2.52628119346454028785751727899, 2.69138847095621900030411632543, 2.96179013381296062116482403623, 3.51799200898076214393234575957, 3.56986651378450047748906429905, 3.98676849454886116192549718051, 4.13113444317770751652418728048, 4.24308495911181245228612306380, 4.61032214450430568109449426950, 4.89955661027194404829139866549, 4.94113669196557769937134841000, 5.19544374056313495832522896435, 5.27088430167978021327512409620, 5.64278476957821505690131161406, 5.84522497395381685060709167758, 5.93303045254508675153672203626, 5.98670371741266532482946664618, 6.40690404072027257896412183166

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