Properties

Degree $8$
Conductor $203928109056$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 8·13-s − 16·25-s − 16·37-s − 2·49-s − 8·61-s + 56·73-s − 5·81-s − 40·97-s + 8·109-s − 16·117-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 2/3·9-s − 2.21·13-s − 3.19·25-s − 2.63·37-s − 2/7·49-s − 1.02·61-s + 6.55·73-s − 5/9·81-s − 4.06·97-s + 0.766·109-s − 1.47·117-s − 0.727·121-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{672} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.119189\)
\(L(\frac12)\) \(\approx\) \(0.119189\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
17$C_2^2$ \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{4} \)
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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

−7.63270450414219960239135295993, −7.25479816294770423799084573660, −7.07984132504543770430997395876, −6.74910554981350092535816366830, −6.73012105073072881703946465272, −6.71117244513687385833961969281, −6.05341147610731636537069524960, −5.77761464814791610263279151894, −5.68120859663521373439814683102, −5.40127792819419315531938345137, −5.19273371515305651271300810021, −4.81431042286693696984655232716, −4.60688793857510330588028211194, −4.58153993771419100457007259389, −3.83731975708371451139929277015, −3.80927786191459989162801470643, −3.77133077181238926323244474587, −3.23365520787058157105723438170, −2.94612821573295632240837478873, −2.38709511202510936307703961529, −2.15295781254606517118379312516, −2.06505240811029814122507348554, −1.62486225674077431030278790305, −1.05578848175629554390963341314, −0.098004205654109376909010208787, 0.098004205654109376909010208787, 1.05578848175629554390963341314, 1.62486225674077431030278790305, 2.06505240811029814122507348554, 2.15295781254606517118379312516, 2.38709511202510936307703961529, 2.94612821573295632240837478873, 3.23365520787058157105723438170, 3.77133077181238926323244474587, 3.80927786191459989162801470643, 3.83731975708371451139929277015, 4.58153993771419100457007259389, 4.60688793857510330588028211194, 4.81431042286693696984655232716, 5.19273371515305651271300810021, 5.40127792819419315531938345137, 5.68120859663521373439814683102, 5.77761464814791610263279151894, 6.05341147610731636537069524960, 6.71117244513687385833961969281, 6.73012105073072881703946465272, 6.74910554981350092535816366830, 7.07984132504543770430997395876, 7.25479816294770423799084573660, 7.63270450414219960239135295993

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