Properties

Label 8-360e4-1.1-c1e4-0-12
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  + 12·13-s + 8·25-s − 32·31-s + 28·37-s + 16·43-s − 48·61-s − 32·67-s − 12·73-s + 20·97-s + 16·103-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 3.32·13-s + 8/5·25-s − 5.74·31-s + 4.60·37-s + 2.43·43-s − 6.14·61-s − 3.90·67-s − 1.40·73-s + 2.03·97-s + 1.57·103-s − 1.81·121-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 + 5.53·169-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 + ⋯

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

\(L(1)\) \(\approx\) \(2.484896876\)
\(L(\frac12)\) \(\approx\) \(2.484896876\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
good7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 158 T^{4} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 1666 T^{4} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 4174 T^{4} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
67$C_2^2$ \( ( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2^3$ \( 1 - 5678 T^{4} + p^{4} T^{8} \)
89$C_2^2$ \( ( 1 + 128 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \)
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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

−8.422086396016260802238858819151, −7.71657750109353381973083640549, −7.69435875814692657276421279218, −7.65984530545156064177098316393, −7.57529067912904237890333773014, −7.15362759833428185811671736880, −6.75428581824896180174500678132, −6.33671650585931222341706671289, −6.15659129300509817029603398526, −5.92740473855369370401405931841, −5.87969600314991506112827772909, −5.65897159369838674556641163117, −5.35289135027668175713423740649, −4.65367612681756547262594712751, −4.44220781751072172516157745919, −4.36745726296670984490516728245, −3.99451668117529367565606951937, −3.70713420445550801069924603530, −3.07147214086244520619584582023, −3.04838393155607177083994596426, −3.00404297095822869316968961654, −1.93513938772314484534875481100, −1.73011782665181828286077374697, −1.35861733944311496550605413841, −0.70539636424688377879002508097, 0.70539636424688377879002508097, 1.35861733944311496550605413841, 1.73011782665181828286077374697, 1.93513938772314484534875481100, 3.00404297095822869316968961654, 3.04838393155607177083994596426, 3.07147214086244520619584582023, 3.70713420445550801069924603530, 3.99451668117529367565606951937, 4.36745726296670984490516728245, 4.44220781751072172516157745919, 4.65367612681756547262594712751, 5.35289135027668175713423740649, 5.65897159369838674556641163117, 5.87969600314991506112827772909, 5.92740473855369370401405931841, 6.15659129300509817029603398526, 6.33671650585931222341706671289, 6.75428581824896180174500678132, 7.15362759833428185811671736880, 7.57529067912904237890333773014, 7.65984530545156064177098316393, 7.69435875814692657276421279218, 7.71657750109353381973083640549, 8.422086396016260802238858819151

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