Properties

Label 8-880e4-1.1-c1e4-0-0
Degree $8$
Conductor $599695360000$
Sign $1$
Analytic cond. $2438.03$
Root an. cond. $2.65081$
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  + 2·3-s + 2·9-s − 18·23-s + 25-s − 4·27-s − 14·37-s + 24·47-s − 12·53-s − 26·67-s − 36·69-s + 12·71-s + 2·75-s − 3·81-s − 34·97-s − 8·103-s − 28·111-s − 42·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 48·141-s + 149-s + 151-s + 157-s − 24·159-s + ⋯
L(s)  = 1  + 1.15·3-s + 2/3·9-s − 3.75·23-s + 1/5·25-s − 0.769·27-s − 2.30·37-s + 3.50·47-s − 1.64·53-s − 3.17·67-s − 4.33·69-s + 1.42·71-s + 0.230·75-s − 1/3·81-s − 3.45·97-s − 0.788·103-s − 2.65·111-s − 3.95·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.04·141-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.90·159-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2438.03\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.6044471887\)
\(L(\frac12)\) \(\approx\) \(0.6044471887\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 - T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \)
7$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$$\times$$C_2^2$ \( ( 1 + 9 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2$$\times$$C_2^2$ \( ( 1 + 7 T + p T^{2} )^{2}( 1 - 25 T^{2} + p^{2} T^{4} ) \)
41$C_2$ \( ( 1 - p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
47$C_2$$\times$$C_2^2$ \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \)
53$C_2$$\times$$C_2^2$ \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )^{2}( 1 + 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - p T^{2} )^{4} \)
67$C_2$$\times$$C_2^2$ \( ( 1 + 13 T + p T^{2} )^{2}( 1 + 35 T^{2} + p^{2} T^{4} ) \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \)
97$C_2$$\times$$C_2^2$ \( ( 1 + 17 T + p T^{2} )^{2}( 1 + 95 T^{2} + p^{2} T^{4} ) \)
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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

−7.25291785836381129923552371752, −7.14121571290375893951754232703, −6.81730525078917028128866720539, −6.64318968508728862006930549438, −6.51994324500531893048001545422, −5.93739024546715926542855212596, −5.89230844967832671849127747393, −5.72005739818956683943062415414, −5.53827942895238510689481172655, −5.25530335625440051681050787391, −4.93211534193629550420975508945, −4.45691717020080695831621076783, −4.22127008656462026332028694621, −4.19427755859513955622835854777, −3.95070026504704050333065034896, −3.64657576234424018936121435033, −3.36457359205956965991956043735, −2.98616379911110058116028989694, −2.87185733507710104143313085673, −2.28126377312779439303269974495, −2.22093221922989517494476629758, −1.78734510435826224394131481270, −1.66189278652108668475544883597, −1.09324360817157726488073739152, −0.16310798549307217222084499513, 0.16310798549307217222084499513, 1.09324360817157726488073739152, 1.66189278652108668475544883597, 1.78734510435826224394131481270, 2.22093221922989517494476629758, 2.28126377312779439303269974495, 2.87185733507710104143313085673, 2.98616379911110058116028989694, 3.36457359205956965991956043735, 3.64657576234424018936121435033, 3.95070026504704050333065034896, 4.19427755859513955622835854777, 4.22127008656462026332028694621, 4.45691717020080695831621076783, 4.93211534193629550420975508945, 5.25530335625440051681050787391, 5.53827942895238510689481172655, 5.72005739818956683943062415414, 5.89230844967832671849127747393, 5.93739024546715926542855212596, 6.51994324500531893048001545422, 6.64318968508728862006930549438, 6.81730525078917028128866720539, 7.14121571290375893951754232703, 7.25291785836381129923552371752

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