Properties

Label 8-882e4-1.1-c2e4-0-8
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  + 2·4-s + 32·13-s + 32·19-s − 32·25-s − 88·31-s + 68·37-s − 160·43-s + 64·52-s − 100·61-s − 8·64-s − 16·67-s + 32·73-s + 64·76-s + 152·79-s + 704·97-s − 64·100-s + 56·103-s − 112·109-s + 46·121-s − 176·124-s + 127-s + 131-s + 137-s + 139-s + 136·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1/2·4-s + 2.46·13-s + 1.68·19-s − 1.27·25-s − 2.83·31-s + 1.83·37-s − 3.72·43-s + 1.23·52-s − 1.63·61-s − 1/8·64-s − 0.238·67-s + 0.438·73-s + 0.842·76-s + 1.92·79-s + 7.25·97-s − 0.639·100-s + 0.543·103-s − 1.02·109-s + 0.380·121-s − 1.41·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.918·148-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.001212948\)
\(L(\frac12)\) \(\approx\) \(3.001212948\)
\(L(2)\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^3$ \( 1 + 32 T^{2} + 399 T^{4} + 32 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2^2$$\times$$C_2^2$ \( ( 1 - 14 T + 75 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )( 1 + 14 T + 75 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} ) \)
13$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{4} \)
17$C_2^3$ \( 1 + 416 T^{2} + 89535 T^{4} + 416 p^{4} T^{6} + p^{8} T^{8} \)
19$C_2^2$ \( ( 1 - 16 T - 105 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 770 T^{2} + 313059 T^{4} + 770 p^{4} T^{6} + p^{8} T^{8} \)
29$C_2^2$ \( ( 1 - 1664 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 + 44 T + 975 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 1184 T^{2} + p^{4} T^{4} )^{2} \)
43$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 2782 T^{2} + 2859843 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \)
53$C_2^3$ \( 1 + 4160 T^{2} + 9415119 T^{4} + 4160 p^{4} T^{6} + p^{8} T^{8} \)
59$C_2^3$ \( 1 + 5810 T^{2} + 21638739 T^{4} + 5810 p^{4} T^{6} + p^{8} T^{8} \)
61$C_2^2$ \( ( 1 + 50 T - 1221 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 8 T - 4425 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 - 7490 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 16 T - 5073 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 76 T - 465 T^{2} - 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 334 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 15680 T^{2} + 183120159 T^{4} + 15680 p^{4} T^{6} + p^{8} T^{8} \)
97$C_2$ \( ( 1 - 176 T + p^{2} T^{2} )^{4} \)
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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

−7.29510597043918513824478881988, −6.68960507114634937417583612715, −6.58454296442981762452584461267, −6.42655846622703584232587513997, −6.08827357587359628402551617166, −5.79926872273733297714545870604, −5.79526518098685410539451085865, −5.70977104975185188391608674850, −5.17620082455211142542757654344, −4.86489606911471324929308752611, −4.86261662107052543545178495424, −4.42394723127628673032919476038, −4.24448905908103766497422397014, −3.67209649549636107384458131348, −3.49185881912232052415658736361, −3.41923875675352544471554605053, −3.25326041481633603663054732158, −3.16231261535100966756269436107, −2.20146259006655249048725598261, −2.08691683668623345723881941881, −2.02502157391267709299706532480, −1.50739936844273005844238514381, −1.11392023832488626151105945597, −0.895555127605426677625014775021, −0.25185809599347690812056891714, 0.25185809599347690812056891714, 0.895555127605426677625014775021, 1.11392023832488626151105945597, 1.50739936844273005844238514381, 2.02502157391267709299706532480, 2.08691683668623345723881941881, 2.20146259006655249048725598261, 3.16231261535100966756269436107, 3.25326041481633603663054732158, 3.41923875675352544471554605053, 3.49185881912232052415658736361, 3.67209649549636107384458131348, 4.24448905908103766497422397014, 4.42394723127628673032919476038, 4.86261662107052543545178495424, 4.86489606911471324929308752611, 5.17620082455211142542757654344, 5.70977104975185188391608674850, 5.79526518098685410539451085865, 5.79926872273733297714545870604, 6.08827357587359628402551617166, 6.42655846622703584232587513997, 6.58454296442981762452584461267, 6.68960507114634937417583612715, 7.29510597043918513824478881988

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