Properties

Label 8-60e8-1.1-c2e4-0-8
Degree $8$
Conductor $1.680\times 10^{14}$
Sign $1$
Analytic cond. $9.25870\times 10^{7}$
Root an. cond. $9.90418$
Motivic weight $2$
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  − 64·19-s − 176·31-s + 164·49-s + 200·61-s − 304·79-s − 92·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 548·169-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  − 3.36·19-s − 5.67·31-s + 3.34·49-s + 3.27·61-s − 3.84·79-s − 0.760·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.24·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(9.25870\times 10^{7}\)
Root analytic conductor: \(9.90418\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 5^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2958976981\)
\(L(\frac12)\) \(\approx\) \(0.2958976981\)
\(L(2)\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2$ \( ( 1 - 14 T + p^{2} T^{2} )^{2}( 1 + 14 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + 416 T^{2} + p^{4} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 770 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1664 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 44 T + p^{2} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 1582 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2098 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 2782 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 + 4160 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 - 50 T + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 8914 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 10402 T^{2} + p^{4} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 76 T + p^{2} T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 334 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 + 12158 T^{2} + p^{4} T^{4} )^{2} \)
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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

−5.69208431744138402385693249313, −5.67073664775078830912228929074, −5.62081713941664506806379264578, −5.31721790804247099144835457599, −5.30072665077767955990513057722, −4.81228236759558101277472382038, −4.51748468176779986932623218938, −4.37292885685795860683821094097, −4.21966178394235150047140077182, −4.00845810303439718205774949314, −3.94155193419132361395351100578, −3.70208875443575326618600155733, −3.33179983992862492140792891550, −3.26201343447385369208709525195, −3.02495812600685257221240920940, −2.47684030591737260637085311734, −2.30777988853573116714202281444, −2.15188237022645097933642209824, −1.98079620785250773521164167292, −1.97235100374910980055079660372, −1.36720093869094286066583849534, −1.21062164738229565714595689696, −0.815549620312793193444007062190, −0.28764384775761840503409300458, −0.10689940642333593782968600210, 0.10689940642333593782968600210, 0.28764384775761840503409300458, 0.815549620312793193444007062190, 1.21062164738229565714595689696, 1.36720093869094286066583849534, 1.97235100374910980055079660372, 1.98079620785250773521164167292, 2.15188237022645097933642209824, 2.30777988853573116714202281444, 2.47684030591737260637085311734, 3.02495812600685257221240920940, 3.26201343447385369208709525195, 3.33179983992862492140792891550, 3.70208875443575326618600155733, 3.94155193419132361395351100578, 4.00845810303439718205774949314, 4.21966178394235150047140077182, 4.37292885685795860683821094097, 4.51748468176779986932623218938, 4.81228236759558101277472382038, 5.30072665077767955990513057722, 5.31721790804247099144835457599, 5.62081713941664506806379264578, 5.67073664775078830912228929074, 5.69208431744138402385693249313

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