Properties

Label 8-300e4-1.1-c1e4-0-2
Degree $8$
Conductor $8100000000$
Sign $1$
Analytic cond. $32.9301$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·31-s + 56·61-s − 9·81-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯
L(s)  = 1  + 2.87·31-s + 7.17·61-s − 81-s + 4·121-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 + 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 + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(32.9301\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.087427694\)
\(L(\frac12)\) \(\approx\) \(2.087427694\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
good7$C_2^3$ \( 1 - 94 T^{4} + p^{4} T^{8} \)
11$C_2$ \( ( 1 - p T^{2} )^{4} \)
13$C_2^3$ \( 1 + 146 T^{4} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 - 2062 T^{4} + p^{4} T^{8} \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^3$ \( 1 - 3214 T^{4} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 + 5906 T^{4} + p^{4} T^{8} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^3$ \( 1 - 8542 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 18814 T^{4} + p^{4} T^{8} \)
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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

−8.395828040338956258713311183158, −8.353456367036923359828498620284, −8.271115255958706346250263979573, −7.86339715024447441911602339562, −7.50798738926167795514168174967, −7.06683226675601598557856900859, −7.03338182723814703896016506530, −6.76857922228371503159190183850, −6.65378478733536299864913578006, −6.02158778135602199514818276050, −5.89180492702692051383416284098, −5.82290054571889066703307141897, −5.29340370255185237460372487794, −5.00657973306111747453348303330, −4.67163413279996298194903414777, −4.62939666683130432817846544659, −3.97975442981480372751068180553, −3.87033357540568871028019178587, −3.51011191800166728322480057009, −3.10809987766291724972668468928, −2.59330614551833375819427227728, −2.34826722357148605880232686547, −2.07065228059845667830178901591, −1.14306251508691973050411521430, −0.799160189443931801462285114043, 0.799160189443931801462285114043, 1.14306251508691973050411521430, 2.07065228059845667830178901591, 2.34826722357148605880232686547, 2.59330614551833375819427227728, 3.10809987766291724972668468928, 3.51011191800166728322480057009, 3.87033357540568871028019178587, 3.97975442981480372751068180553, 4.62939666683130432817846544659, 4.67163413279996298194903414777, 5.00657973306111747453348303330, 5.29340370255185237460372487794, 5.82290054571889066703307141897, 5.89180492702692051383416284098, 6.02158778135602199514818276050, 6.65378478733536299864913578006, 6.76857922228371503159190183850, 7.03338182723814703896016506530, 7.06683226675601598557856900859, 7.50798738926167795514168174967, 7.86339715024447441911602339562, 8.271115255958706346250263979573, 8.353456367036923359828498620284, 8.395828040338956258713311183158

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