Properties

Label 8-1150e4-1.1-c1e4-0-0
Degree $8$
Conductor $1.749\times 10^{12}$
Sign $1$
Analytic cond. $7110.49$
Root an. cond. $3.03031$
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 + 9-s − 14·11-s + 3·16-s − 2·19-s − 4·29-s − 10·31-s − 2·36-s − 18·41-s + 28·44-s + 17·49-s + 28·59-s + 10·61-s − 4·64-s − 58·71-s + 4·76-s + 12·81-s − 14·99-s − 4·101-s − 42·109-s + 8·116-s + 85·121-s + 20·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 4-s + 1/3·9-s − 4.22·11-s + 3/4·16-s − 0.458·19-s − 0.742·29-s − 1.79·31-s − 1/3·36-s − 2.81·41-s + 4.22·44-s + 17/7·49-s + 3.64·59-s + 1.28·61-s − 1/2·64-s − 6.88·71-s + 0.458·76-s + 4/3·81-s − 1.40·99-s − 0.398·101-s − 4.02·109-s + 0.742·116-s + 7.72·121-s + 1.79·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 5^{8} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(7110.49\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 5^{8} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2547357215\)
\(L(\frac12)\) \(\approx\) \(0.2547357215\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T^{2} )^{2} \)
5 \( 1 \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$C_4\times C_2$ \( 1 - T^{2} - 11 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
7$D_4\times C_2$ \( 1 - 17 T^{2} + 141 T^{4} - 17 p^{2} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 41 T^{2} + 729 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 5 T^{2} + 321 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 60 T^{2} + 3766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 160 T^{2} + 10766 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 76 T^{2} + 3734 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 29 T + 349 T^{2} + 29 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 8 T^{2} - 1026 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 196 T^{2} + 20054 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$D_4\times C_2$ \( 1 - 341 T^{2} + 47625 T^{4} - 341 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.05069534485786093392430640588, −6.84461407946393813270972277500, −6.73309595043211026452635701941, −6.39312512637828103682891808541, −5.74773463783686144049757919619, −5.66066747521837823662955915601, −5.62232521436253832620673770507, −5.47567020337299612441829422783, −5.35727916230254136361379841954, −5.04265455870099342961153824576, −4.66665806121396910012215287151, −4.56887612037621709218770016165, −4.43874567944835561252862202381, −3.81331076608123693239281415861, −3.69966725540277802712283340580, −3.68607717001711619305555499634, −3.18433928996830021773184473887, −2.72714639332752488043557049530, −2.58828813060652595344920831582, −2.53431098249378248139891989219, −2.10826978460110325798968108821, −1.59339858275510356301425609484, −1.43083537192613349746032026409, −0.51602776230605901476758683100, −0.18563452770645763297984841007, 0.18563452770645763297984841007, 0.51602776230605901476758683100, 1.43083537192613349746032026409, 1.59339858275510356301425609484, 2.10826978460110325798968108821, 2.53431098249378248139891989219, 2.58828813060652595344920831582, 2.72714639332752488043557049530, 3.18433928996830021773184473887, 3.68607717001711619305555499634, 3.69966725540277802712283340580, 3.81331076608123693239281415861, 4.43874567944835561252862202381, 4.56887612037621709218770016165, 4.66665806121396910012215287151, 5.04265455870099342961153824576, 5.35727916230254136361379841954, 5.47567020337299612441829422783, 5.62232521436253832620673770507, 5.66066747521837823662955915601, 5.74773463783686144049757919619, 6.39312512637828103682891808541, 6.73309595043211026452635701941, 6.84461407946393813270972277500, 7.05069534485786093392430640588

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