Properties

Label 8-570e4-1.1-c1e4-0-6
Degree $8$
Conductor $105560010000$
Sign $1$
Analytic cond. $429.148$
Root an. cond. $2.13341$
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  − 2·4-s + 2·9-s + 3·16-s + 4·19-s + 8·25-s + 24·29-s − 4·36-s − 24·41-s + 28·49-s + 24·59-s + 40·61-s − 4·64-s − 48·71-s − 8·76-s − 5·81-s + 48·89-s − 16·100-s − 48·116-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 6·144-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  − 4-s + 2/3·9-s + 3/4·16-s + 0.917·19-s + 8/5·25-s + 4.45·29-s − 2/3·36-s − 3.74·41-s + 4·49-s + 3.12·59-s + 5.12·61-s − 1/2·64-s − 5.69·71-s − 0.917·76-s − 5/9·81-s + 5.08·89-s − 8/5·100-s − 4.45·116-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(429.148\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.904797835\)
\(L(\frac12)\) \(\approx\) \(2.904797835\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
good7$C_2$ \( ( 1 - p T^{2} )^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2^2$ \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
31$C_2^2$ \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
43$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{4} \)
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

−7.952495216965527980040992332389, −7.22427380538516455557759379021, −7.14017456933358778711236133644, −7.08069703236494460988782277530, −6.89736911264218578276863381223, −6.82711209050152769277170487909, −6.16912074165678830344209839632, −6.07379570726628005858886730886, −5.83691385684202648853490063127, −5.45607780593449915246436642881, −5.19207214547189710149614703575, −4.86533141844064791464779038261, −4.78760870055364972844170449923, −4.71888218608363535745483737791, −4.16462087170743211196399345504, −4.02413681875158772330254183995, −3.62176628450702958439123385199, −3.28287694755221768717613138887, −3.21558569146028422615279541221, −2.61109188884886137685950101304, −2.38954943759360022993204738220, −2.10781613632974988767671511826, −1.17081022231076634816454056399, −1.03545817189919755906969914572, −0.73166274290232916302434910342, 0.73166274290232916302434910342, 1.03545817189919755906969914572, 1.17081022231076634816454056399, 2.10781613632974988767671511826, 2.38954943759360022993204738220, 2.61109188884886137685950101304, 3.21558569146028422615279541221, 3.28287694755221768717613138887, 3.62176628450702958439123385199, 4.02413681875158772330254183995, 4.16462087170743211196399345504, 4.71888218608363535745483737791, 4.78760870055364972844170449923, 4.86533141844064791464779038261, 5.19207214547189710149614703575, 5.45607780593449915246436642881, 5.83691385684202648853490063127, 6.07379570726628005858886730886, 6.16912074165678830344209839632, 6.82711209050152769277170487909, 6.89736911264218578276863381223, 7.08069703236494460988782277530, 7.14017456933358778711236133644, 7.22427380538516455557759379021, 7.952495216965527980040992332389

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