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Properties

Label 8-75e4-1.1-c4e4-0-4
Degree $8$
Conductor $31640625$
Sign $1$
Analytic cond. $3612.62$
Root an. cond. $2.78437$
Motivic weight $4$
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  + 44·4-s + 72·9-s + 940·16-s − 1.23e3·19-s + 128·31-s + 3.16e3·36-s − 9.13e3·49-s − 3.71e3·61-s + 8.80e3·64-s − 5.42e4·76-s − 32·79-s − 1.37e3·81-s + 2.54e4·109-s + 3.98e4·121-s + 5.63e3·124-s + 127-s + 131-s + 137-s + 139-s + 6.76e4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.56e3·169-s − 8.87e4·171-s + ⋯
L(s)  = 1  + 11/4·4-s + 8/9·9-s + 3.67·16-s − 3.41·19-s + 0.133·31-s + 22/9·36-s − 3.80·49-s − 0.997·61-s + 2.14·64-s − 9.38·76-s − 0.00512·79-s − 0.209·81-s + 2.13·109-s + 2.72·121-s + 0.366·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 3.26·144-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 0.194·169-s − 3.03·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(31640625\)    =    \(3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(3612.62\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 31640625,\ (\ :2, 2, 2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.016314517\)
\(L(\frac12)\) \(\approx\) \(5.016314517\)
\(L(3)\) 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)$
bad3$C_2^2$ \( 1 - 8 p^{2} T^{2} + p^{8} T^{4} \)
5 \( 1 \)
good2$C_2^2$ \( ( 1 - 11 p T^{2} + p^{8} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 + 4568 T^{2} + p^{8} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 19922 T^{2} + p^{8} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 214 p T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 89602 T^{2} + p^{8} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 308 T + p^{4} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 - 388072 T^{2} + p^{8} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1377122 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 32 T + p^{4} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 2097218 T^{2} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 1324862 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 154328 T^{2} + p^{8} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3784072 T^{2} + p^{8} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 13773922 T^{2} + p^{8} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 8500562 T^{2} + p^{8} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 928 T + p^{4} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 33618968 T^{2} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 29257922 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 38971298 T^{2} + p^{8} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 8 T + p^{4} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 75232552 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 39222722 T^{2} + p^{8} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 170764898 T^{2} + p^{8} T^{4} )^{2} \)
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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

−10.25949964943031990456708898712, −9.863971345531952065343425164016, −9.369103140582373527845874025808, −9.290404020332520169555084294406, −8.674978203427884348229391751629, −8.194938513935151130105712395360, −8.131128173456404989091139422749, −7.968936007023284562318892157503, −7.21248292970861316165984481824, −7.06132179658095188282352393732, −6.82944989024557793775279229477, −6.68107233291403665662742846401, −6.10577109875096889924542365741, −6.00656225657649521388237613816, −5.83908078730190760662338582089, −4.83301708454504493064409526956, −4.53526844587481686923438187718, −4.39555455396191878173550644546, −3.57009961599535403304284743719, −3.17000569511315101721430677944, −2.74068181530285147226818461064, −2.09357639942449417823164059186, −1.74114622374841319534927830500, −1.73793635366936271801659769751, −0.46398618199337959240624110096, 0.46398618199337959240624110096, 1.73793635366936271801659769751, 1.74114622374841319534927830500, 2.09357639942449417823164059186, 2.74068181530285147226818461064, 3.17000569511315101721430677944, 3.57009961599535403304284743719, 4.39555455396191878173550644546, 4.53526844587481686923438187718, 4.83301708454504493064409526956, 5.83908078730190760662338582089, 6.00656225657649521388237613816, 6.10577109875096889924542365741, 6.68107233291403665662742846401, 6.82944989024557793775279229477, 7.06132179658095188282352393732, 7.21248292970861316165984481824, 7.968936007023284562318892157503, 8.131128173456404989091139422749, 8.194938513935151130105712395360, 8.674978203427884348229391751629, 9.290404020332520169555084294406, 9.369103140582373527845874025808, 9.863971345531952065343425164016, 10.25949964943031990456708898712

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