Properties

Label 8-1440e4-1.1-c1e4-0-16
Degree $8$
Conductor $4.300\times 10^{12}$
Sign $1$
Analytic cond. $17480.6$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·25-s − 24·29-s + 24·89-s + 72·101-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 2·25-s − 4.45·29-s + 2.54·89-s + 7.16·101-s − 4·121-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 + 4·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{20} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(17480.6\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1440} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{20} \cdot 3^{8} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.591610151\)
\(L(\frac12)\) \(\approx\) \(2.591610151\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T^{2} )^{2} \)
good7$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{4} \)
13$C_2$ \( ( 1 - p T^{2} )^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$$\times$$C_2^2$ \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{4} \)
41$C_2^2$ \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2^2$$\times$$C_2^2$ \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \)
47$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \)
53$C_2$ \( ( 1 - p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \)
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_2^2$$\times$$C_2^2$ \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
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

−6.83244411405491439020095373372, −6.55526190724563738863186979149, −6.23787131725952708877832366772, −6.22429558371732792525081163101, −6.05787805785935546198358658898, −5.57816554126748423400448726305, −5.54407433023342189273055146191, −5.13790483574158477424427236815, −5.06810304912545001160237796950, −5.03719330713400564365034910379, −4.52923907614230466791568109167, −4.26982624030651213656448852857, −4.16114513807988639728471297203, −3.78889472190645049311628637546, −3.52220768032087563219414739347, −3.31906518039096453568612848383, −3.26955170548487771005438957371, −2.82538235659231648077356102088, −2.45767430073303107360858235149, −2.12072966994198831819881376486, −1.92637502767769634421162904253, −1.72793885768563478206225408784, −1.23629850438442708890282226677, −0.72238767160405768456231364357, −0.39156806227144678853467505921, 0.39156806227144678853467505921, 0.72238767160405768456231364357, 1.23629850438442708890282226677, 1.72793885768563478206225408784, 1.92637502767769634421162904253, 2.12072966994198831819881376486, 2.45767430073303107360858235149, 2.82538235659231648077356102088, 3.26955170548487771005438957371, 3.31906518039096453568612848383, 3.52220768032087563219414739347, 3.78889472190645049311628637546, 4.16114513807988639728471297203, 4.26982624030651213656448852857, 4.52923907614230466791568109167, 5.03719330713400564365034910379, 5.06810304912545001160237796950, 5.13790483574158477424427236815, 5.54407433023342189273055146191, 5.57816554126748423400448726305, 6.05787805785935546198358658898, 6.22429558371732792525081163101, 6.23787131725952708877832366772, 6.55526190724563738863186979149, 6.83244411405491439020095373372

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