Properties

Label 8-570e4-1.1-c1e4-0-7
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  + 4·2-s + 4·3-s + 10·4-s + 16·6-s + 8·7-s + 20·8-s + 6·9-s + 40·12-s + 32·14-s + 35·16-s + 24·18-s + 32·21-s + 80·24-s − 2·25-s − 4·27-s + 80·28-s − 8·29-s + 56·32-s + 60·36-s − 16·41-s + 128·42-s + 32·43-s + 140·48-s + 16·49-s − 8·50-s − 24·53-s − 16·54-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 6.53·6-s + 3.02·7-s + 7.07·8-s + 2·9-s + 11.5·12-s + 8.55·14-s + 35/4·16-s + 5.65·18-s + 6.98·21-s + 16.3·24-s − 2/5·25-s − 0.769·27-s + 15.1·28-s − 1.48·29-s + 9.89·32-s + 10·36-s − 2.49·41-s + 19.7·42-s + 4.87·43-s + 20.2·48-s + 16/7·49-s − 1.13·50-s − 3.29·53-s − 2.17·54-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

\(L(1)\) \(\approx\) \(60.28216530\)
\(L(\frac12)\) \(\approx\) \(60.28216530\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
good7$D_{4}$ \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 8 T^{2} + 130 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 104 T^{2} + 5154 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 - 16 T + 148 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 100 T^{2} + 5766 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{4} \)
71$C_2^2$ \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 52 T^{2} + 11110 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 116 T^{2} + 13942 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 256 T^{2} + 34690 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.80202242227694047962297532147, −7.60048286717238886568321299370, −7.41776250938688509094261352850, −7.05893811535262297355140259962, −6.93632721454390896411354400547, −6.15120139300849228076225475644, −6.13802179362875063779122290392, −6.07107195804993533841591465537, −5.82422646652259103790099859648, −5.33073275625658685155226248282, −5.13496602190129717167897367182, −4.83505278822222212966969056197, −4.72734722309184612335936219697, −4.48588244476838388878468021497, −4.19930328807414538843215952970, −4.06691870951529108305198035204, −3.51427677932967976688097641075, −3.29687294806212818937218886176, −3.19322445155388735948479048089, −2.94739214611423107813885716974, −2.39148028283665903808328174644, −2.02476914482624975678854849834, −1.98662185733131327668353988066, −1.62043900200395382227800313105, −1.39619237750898003875282140663, 1.39619237750898003875282140663, 1.62043900200395382227800313105, 1.98662185733131327668353988066, 2.02476914482624975678854849834, 2.39148028283665903808328174644, 2.94739214611423107813885716974, 3.19322445155388735948479048089, 3.29687294806212818937218886176, 3.51427677932967976688097641075, 4.06691870951529108305198035204, 4.19930328807414538843215952970, 4.48588244476838388878468021497, 4.72734722309184612335936219697, 4.83505278822222212966969056197, 5.13496602190129717167897367182, 5.33073275625658685155226248282, 5.82422646652259103790099859648, 6.07107195804993533841591465537, 6.13802179362875063779122290392, 6.15120139300849228076225475644, 6.93632721454390896411354400547, 7.05893811535262297355140259962, 7.41776250938688509094261352850, 7.60048286717238886568321299370, 7.80202242227694047962297532147

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