Properties

Label 8-360e4-1.1-c1e4-0-1
Degree $8$
Conductor $16796160000$
Sign $1$
Analytic cond. $68.2839$
Root an. cond. $1.69546$
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  − 4·4-s + 4·7-s − 8·13-s + 12·16-s + 8·25-s − 16·28-s − 32·37-s − 8·43-s + 8·49-s + 32·52-s − 32·64-s − 32·67-s − 20·73-s + 56·79-s − 32·91-s + 28·97-s − 32·100-s − 20·103-s − 16·109-s + 48·112-s − 8·121-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + ⋯
L(s)  = 1  − 2·4-s + 1.51·7-s − 2.21·13-s + 3·16-s + 8/5·25-s − 3.02·28-s − 5.26·37-s − 1.21·43-s + 8/7·49-s + 4.43·52-s − 4·64-s − 3.90·67-s − 2.34·73-s + 6.30·79-s − 3.35·91-s + 2.84·97-s − 3.19·100-s − 1.97·103-s − 1.53·109-s + 4.53·112-s − 0.727·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 10.5·148-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(68.2839\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4578115242\)
\(L(\frac12)\) \(\approx\) \(0.4578115242\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
3 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
good7$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 - 254 T^{4} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 706 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 3682 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 4786 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{4} \)
83$C_2^3$ \( 1 + 12466 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−8.237556966041645477872060617016, −8.155946950454701898944276474150, −7.81032563822009154568448863843, −7.57622252429694890301132363395, −7.56007511873165763545427149862, −7.08598655242055770511521296733, −6.78458012392423622974426439809, −6.69482067536313519997633694233, −6.27998122146795823254546000414, −5.82424472434253072344866945587, −5.44026497864630562994183716041, −5.15898382064575334971704120403, −5.05918100560641238842244979596, −5.00644612604560386133816369525, −4.75385983124558398975947220182, −4.46400931047827915708177585734, −3.99511459055492335677291937081, −3.77112795772153319554191979550, −3.32556572233222009771310441535, −3.12347640334645677877465977531, −2.67789938716730651264373725699, −2.05992231650708133928677586606, −1.64285982951218278896048291358, −1.33975368803360284147779574736, −0.30020967791384150292187489571, 0.30020967791384150292187489571, 1.33975368803360284147779574736, 1.64285982951218278896048291358, 2.05992231650708133928677586606, 2.67789938716730651264373725699, 3.12347640334645677877465977531, 3.32556572233222009771310441535, 3.77112795772153319554191979550, 3.99511459055492335677291937081, 4.46400931047827915708177585734, 4.75385983124558398975947220182, 5.00644612604560386133816369525, 5.05918100560641238842244979596, 5.15898382064575334971704120403, 5.44026497864630562994183716041, 5.82424472434253072344866945587, 6.27998122146795823254546000414, 6.69482067536313519997633694233, 6.78458012392423622974426439809, 7.08598655242055770511521296733, 7.56007511873165763545427149862, 7.57622252429694890301132363395, 7.81032563822009154568448863843, 8.155946950454701898944276474150, 8.237556966041645477872060617016

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