Properties

Label 8-475e4-1.1-c2e4-0-0
Degree $8$
Conductor $50906640625$
Sign $1$
Analytic cond. $28061.7$
Root an. cond. $3.59761$
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  + 10·4-s − 10·9-s − 40·11-s + 43·16-s + 24·19-s − 100·36-s − 400·44-s + 146·49-s − 160·61-s + 20·64-s + 240·76-s − 87·81-s + 400·99-s + 516·121-s + 127-s + 131-s + 137-s + 139-s − 430·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 650·169-s − 240·171-s + 173-s + ⋯
L(s)  = 1  + 5/2·4-s − 1.11·9-s − 3.63·11-s + 2.68·16-s + 1.26·19-s − 2.77·36-s − 9.09·44-s + 2.97·49-s − 2.62·61-s + 5/16·64-s + 3.15·76-s − 1.07·81-s + 4.04·99-s + 4.26·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 2.98·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.84·169-s − 1.40·171-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(28061.7\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{8} \cdot 19^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01232611276\)
\(L(\frac12)\) \(\approx\) \(0.01232611276\)
\(L(2)\) 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)$
bad5 \( 1 \)
19$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 - 5 T^{2} + p^{4} T^{4} )^{2} \)
3$C_2^2$ \( ( 1 + 5 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 73 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{4} \)
13$C_2^2$ \( ( 1 + 25 p T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 353 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 167 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1357 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 622 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 2270 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 3298 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 4318 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 115 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 6637 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 7405 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2$ \( ( 1 - 92 T + p^{2} T^{2} )^{2}( 1 + 92 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( ( 1 + 367 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 11182 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 12178 T^{2} + p^{4} T^{4} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{4}( 1 + p T )^{4} \)
97$C_2^2$ \( ( 1 + 3790 T^{2} + p^{4} T^{4} )^{2} \)
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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

−7.80822325794979319846646500452, −7.68843015395709665593182984573, −7.10306209226862497409887226946, −7.00459733056120416094037082151, −7.00410662989596097938371708543, −6.64251629629239723213848085928, −5.99123162871928463389035959198, −5.90431373673978617743100352714, −5.87789743029339185414523297867, −5.72166928778086509262633407036, −5.23139667101396254989610206503, −5.08907327622472302920306048483, −4.80724642784834275126783989004, −4.57106977885435652762082776236, −3.84963676997206974078602232385, −3.81087305377229716940895857559, −3.04738396981920989676950008915, −2.96031240123102915409498675116, −2.80899820967120566905373291070, −2.60000399363871039369402658868, −2.30818173398764813827976122636, −1.99153945819978947106213986472, −1.47860304651263597695840423328, −0.952712733363147178738686937538, −0.01869733813459852836241354637, 0.01869733813459852836241354637, 0.952712733363147178738686937538, 1.47860304651263597695840423328, 1.99153945819978947106213986472, 2.30818173398764813827976122636, 2.60000399363871039369402658868, 2.80899820967120566905373291070, 2.96031240123102915409498675116, 3.04738396981920989676950008915, 3.81087305377229716940895857559, 3.84963676997206974078602232385, 4.57106977885435652762082776236, 4.80724642784834275126783989004, 5.08907327622472302920306048483, 5.23139667101396254989610206503, 5.72166928778086509262633407036, 5.87789743029339185414523297867, 5.90431373673978617743100352714, 5.99123162871928463389035959198, 6.64251629629239723213848085928, 7.00410662989596097938371708543, 7.00459733056120416094037082151, 7.10306209226862497409887226946, 7.68843015395709665593182984573, 7.80822325794979319846646500452

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