Properties

Label 8-360e4-1.1-c1e4-0-0
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 − 8·7-s + 16·13-s + 12·16-s − 10·25-s + 32·28-s − 32·37-s + 32·49-s − 64·52-s − 32·64-s − 128·91-s + 40·100-s − 56·103-s − 96·112-s + 127-s + 131-s + 137-s + 139-s + 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 128·169-s + 173-s + 80·175-s + ⋯
L(s)  = 1  − 2·4-s − 3.02·7-s + 4.43·13-s + 3·16-s − 2·25-s + 6.04·28-s − 5.26·37-s + 32/7·49-s − 8.87·52-s − 4·64-s − 13.4·91-s + 4·100-s − 5.51·103-s − 9.07·112-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 + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 9.84·169-s + 0.0760·173-s + 6.04·175-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: induced by $\chi_{360} (1, \cdot )$
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

\(L(1)\) \(\approx\) \(0.1339312164\)
\(L(\frac12)\) \(\approx\) \(0.1339312164\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 82 T^{4} + p^{4} T^{8} \)
13$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 + 2722 T^{4} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - p T^{2} )^{4} \)
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$C_2^3$ \( 1 - 878 T^{4} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - p T^{2} )^{4} \)
79$C_2$ \( ( 1 - p T^{2} )^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2^3$ \( 1 - 15518 T^{4} + p^{4} T^{8} \)
97$C_2$ \( ( 1 - p T^{2} )^{4} \)
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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

−8.542224948287722447535287943551, −8.233526803367150988863226753917, −7.85783261441437572162572179951, −7.85652118160776739448918885789, −6.96579269380390658844727598228, −6.95314543650627582541603963578, −6.88331381211836733152531048145, −6.43774406431362764392197294121, −6.26264692259677652491296357441, −5.79932049143313514064481845277, −5.70750720380320135461163522127, −5.68511512222194598575690538820, −5.41382295049435944719099842825, −4.80588587888937454904143072103, −4.51346266939693088869012836216, −3.89100685333559746365301111849, −3.86270586516254320614313577071, −3.63710201170448765273606466771, −3.58618667309477370457527973925, −3.17269742691337179327917510861, −3.08737558960831532393239133987, −2.15386693707151026946802445022, −1.36474661679798379689652479566, −1.31346534934583332374835228493, −0.17623747988585091616316784497, 0.17623747988585091616316784497, 1.31346534934583332374835228493, 1.36474661679798379689652479566, 2.15386693707151026946802445022, 3.08737558960831532393239133987, 3.17269742691337179327917510861, 3.58618667309477370457527973925, 3.63710201170448765273606466771, 3.86270586516254320614313577071, 3.89100685333559746365301111849, 4.51346266939693088869012836216, 4.80588587888937454904143072103, 5.41382295049435944719099842825, 5.68511512222194598575690538820, 5.70750720380320135461163522127, 5.79932049143313514064481845277, 6.26264692259677652491296357441, 6.43774406431362764392197294121, 6.88331381211836733152531048145, 6.95314543650627582541603963578, 6.96579269380390658844727598228, 7.85652118160776739448918885789, 7.85783261441437572162572179951, 8.233526803367150988863226753917, 8.542224948287722447535287943551

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