Properties

Label 8-507e4-1.1-c1e4-0-4
Degree $8$
Conductor $66074188401$
Sign $1$
Analytic cond. $268.621$
Root an. cond. $2.01206$
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  + 2·3-s + 4-s + 9-s + 2·12-s + 4·16-s − 12·17-s − 20·25-s − 2·27-s − 12·29-s + 36-s − 8·43-s + 8·48-s + 2·49-s − 24·51-s + 24·53-s + 4·61-s + 11·64-s − 12·68-s − 40·75-s − 32·79-s − 4·81-s − 24·87-s − 20·100-s + 12·101-s − 32·103-s − 24·107-s − 2·108-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s + 1/3·9-s + 0.577·12-s + 16-s − 2.91·17-s − 4·25-s − 0.384·27-s − 2.22·29-s + 1/6·36-s − 1.21·43-s + 1.15·48-s + 2/7·49-s − 3.36·51-s + 3.29·53-s + 0.512·61-s + 11/8·64-s − 1.45·68-s − 4.61·75-s − 3.60·79-s − 4/9·81-s − 2.57·87-s − 2·100-s + 1.19·101-s − 3.15·103-s − 2.32·107-s − 0.192·108-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(268.621\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{507} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 13^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.9838901723\)
\(L(\frac12)\) \(\approx\) \(0.9838901723\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( ( 1 - T + T^{2} )^{2} \)
13 \( 1 \)
good2$C_2^3$ \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
5$C_2$ \( ( 1 + p T^{2} )^{4} \)
7$C_2^2$$\times$$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
11$C_2^3$ \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2$ \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
19$C_2^2$$\times$$C_2^2$ \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \)
23$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \)
37$C_2^2$$\times$$C_2^2$ \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \)
41$C_2^3$ \( 1 - 34 T^{2} - 525 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 - 10 T^{2} - 3381 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 26 T^{2} - 3813 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^3$ \( 1 - 130 T^{2} + 11859 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2$ \( ( 1 + p T^{2} )^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 + 154 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 - 130 T^{2} + 8979 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$$\times$$C_2^2$ \( ( 1 - 169 T^{2} + p^{2} T^{4} )( 1 + 167 T^{2} + p^{2} T^{4} ) \)
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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

−8.105864610898901813475340139241, −7.54680014348575957132171804626, −7.40846100464523643984076280573, −7.27017958399500349143551862434, −7.06214387103409971680685012792, −6.60014161382903242036073866970, −6.57274609028942619329965000870, −6.21434428146745364696109689731, −5.78741246628291012785137877972, −5.71132870315572181055302443724, −5.47761857973137237889707746894, −5.30381557664612946007544395456, −4.85090603086031049925575186469, −4.33232017353456202698221065040, −4.02311220813120766645505299471, −3.96586037906396517021802940317, −3.88752404187981442698898770881, −3.52649651814239581667528243385, −2.95834841522328463941104787985, −2.58674421072034485304408973566, −2.47213258723872095346986646651, −2.00196344104988349130777804531, −1.82202202495074669595275408454, −1.51288112759546948772989045946, −0.26348764836235493063783754692, 0.26348764836235493063783754692, 1.51288112759546948772989045946, 1.82202202495074669595275408454, 2.00196344104988349130777804531, 2.47213258723872095346986646651, 2.58674421072034485304408973566, 2.95834841522328463941104787985, 3.52649651814239581667528243385, 3.88752404187981442698898770881, 3.96586037906396517021802940317, 4.02311220813120766645505299471, 4.33232017353456202698221065040, 4.85090603086031049925575186469, 5.30381557664612946007544395456, 5.47761857973137237889707746894, 5.71132870315572181055302443724, 5.78741246628291012785137877972, 6.21434428146745364696109689731, 6.57274609028942619329965000870, 6.60014161382903242036073866970, 7.06214387103409971680685012792, 7.27017958399500349143551862434, 7.40846100464523643984076280573, 7.54680014348575957132171804626, 8.105864610898901813475340139241

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