Properties

Label 8-825e4-1.1-c1e4-0-11
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7·16-s + 24·29-s + 16·31-s + 8·61-s − 9·81-s − 48·89-s − 2·121-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 + ⋯
L(s)  = 1  + 7/4·16-s + 4.45·29-s + 2.87·31-s + 1.02·61-s − 81-s − 5.08·89-s − 0.181·121-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(4.487561598\)
\(L(\frac12)\) \(\approx\) \(4.487561598\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2^2$ \( 1 + p^{2} T^{4} \)
5 \( 1 \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
good2$C_2^3$ \( 1 - 7 T^{4} + p^{4} T^{8} \)
7$C_2^3$ \( 1 - 94 T^{4} + p^{4} T^{8} \)
13$C_2^3$ \( 1 - 142 T^{4} + p^{4} T^{8} \)
17$C_2^3$ \( 1 - 382 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 1054 T^{4} + p^{4} T^{8} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 3214 T^{4} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 2302 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 2254 T^{4} + p^{4} T^{8} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
67$C_2^3$ \( 1 + 5906 T^{4} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 7298 T^{4} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 10414 T^{4} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 18814 T^{4} + p^{4} T^{8} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.31397505138621098325041298662, −7.12103515215724364093797038254, −6.82666626883813141003240335385, −6.57353562060503103428983826745, −6.35342702350562535013859390636, −6.24854159547523000432739262837, −6.23206132091284541611808403328, −5.48163109173721446746926093859, −5.48082224056076259347807263938, −5.37112422912152037541155954165, −5.06622347074096381439117967567, −4.55857720227933559484928522294, −4.38556169936754647212739060691, −4.35099814839084804542718503123, −4.12988969494698604803712393759, −3.67210192696209675670644753896, −3.12292974216866090908312791472, −3.07371749957662860629297308751, −2.86193901915992400633092457262, −2.64628511012115663194687078614, −2.31535271266320038460972321147, −1.54326258745598864247748837231, −1.43094132361273501017526129662, −0.831867587077564432072177027364, −0.73640372214755439074320240339, 0.73640372214755439074320240339, 0.831867587077564432072177027364, 1.43094132361273501017526129662, 1.54326258745598864247748837231, 2.31535271266320038460972321147, 2.64628511012115663194687078614, 2.86193901915992400633092457262, 3.07371749957662860629297308751, 3.12292974216866090908312791472, 3.67210192696209675670644753896, 4.12988969494698604803712393759, 4.35099814839084804542718503123, 4.38556169936754647212739060691, 4.55857720227933559484928522294, 5.06622347074096381439117967567, 5.37112422912152037541155954165, 5.48082224056076259347807263938, 5.48163109173721446746926093859, 6.23206132091284541611808403328, 6.24854159547523000432739262837, 6.35342702350562535013859390636, 6.57353562060503103428983826745, 6.82666626883813141003240335385, 7.12103515215724364093797038254, 7.31397505138621098325041298662

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