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Properties

Label 8-1875e4-1.1-c1e4-0-4
Degree $8$
Conductor $1.236\times 10^{13}$
Sign $1$
Analytic cond. $50247.3$
Root an. cond. $3.86936$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4·3-s − 3·4-s − 4·6-s + 5·7-s − 5·8-s + 10·9-s − 6·11-s + 12·12-s + 7·13-s + 5·14-s − 7·17-s + 10·18-s − 9·19-s − 20·21-s − 6·22-s + 10·23-s + 20·24-s + 7·26-s − 20·27-s − 15·28-s − 28·29-s − 10·31-s + 9·32-s + 24·33-s − 7·34-s − 30·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 2.30·3-s − 3/2·4-s − 1.63·6-s + 1.88·7-s − 1.76·8-s + 10/3·9-s − 1.80·11-s + 3.46·12-s + 1.94·13-s + 1.33·14-s − 1.69·17-s + 2.35·18-s − 2.06·19-s − 4.36·21-s − 1.27·22-s + 2.08·23-s + 4.08·24-s + 1.37·26-s − 3.84·27-s − 2.83·28-s − 5.19·29-s − 1.79·31-s + 1.59·32-s + 4.17·33-s − 1.20·34-s − 5·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(50247.3\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5 \( 1 \)
good2$C_4\times C_2$ \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 - 5 T + 4 p T^{2} - 95 T^{3} + 289 T^{4} - 95 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 + 6 T + 50 T^{2} + 189 T^{3} + 849 T^{4} + 189 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 46 T^{2} - 181 T^{3} + 769 T^{4} - 181 p T^{5} + 46 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 + 7 T + 52 T^{2} + 155 T^{3} + 831 T^{4} + 155 p T^{5} + 52 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 82 T^{2} + 477 T^{3} + 2385 T^{4} + 477 p T^{5} + 82 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 112 T^{2} - 680 T^{3} + 4089 T^{4} - 680 p T^{5} + 112 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 28 T + 400 T^{2} + 3653 T^{3} + 23319 T^{4} + 3653 p T^{5} + 400 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 4 p T^{2} + 805 T^{3} + 5641 T^{4} + 805 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 58 T^{2} + 85 T^{3} + 79 T^{4} + 85 p T^{5} + 58 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 94 T^{2} - 135 T^{3} + 4491 T^{4} - 135 p T^{5} + 94 p^{2} T^{6} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - T + 93 T^{2} - 470 T^{3} + 3881 T^{4} - 470 p T^{5} + 93 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 23 T + 277 T^{2} + 2230 T^{3} + 15531 T^{4} + 2230 p T^{5} + 277 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 67 T^{2} - 270 T^{3} + 4479 T^{4} - 270 p T^{5} + 67 p^{2} T^{6} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 82 T^{2} - 287 T^{3} + 4245 T^{4} - 287 p T^{5} + 82 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 43 T + 913 T^{2} + 12286 T^{3} + 114205 T^{4} + 12286 p T^{5} + 913 p^{2} T^{6} + 43 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 207 T^{2} - 1150 T^{3} + 18911 T^{4} - 1150 p T^{5} + 207 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 27 T + 418 T^{2} + 5079 T^{3} + 49545 T^{4} + 5079 p T^{5} + 418 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 232 T^{2} - 2385 T^{3} + 25959 T^{4} - 2385 p T^{5} + 232 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 - 10 T + 211 T^{2} - 1000 T^{3} + 17701 T^{4} - 1000 p T^{5} + 211 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 - 3 T + 86 T^{2} - 549 T^{3} + 13989 T^{4} - 549 p T^{5} + 86 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 9 T + 352 T^{2} + 2307 T^{3} + 46875 T^{4} + 2307 p T^{5} + 352 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 172 T^{2} - 2335 T^{3} + 29251 T^{4} - 2335 p T^{5} + 172 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.83485545036000741627496619751, −6.78095581366020259871362525636, −6.31895140748792489154506104026, −6.28275183223450232246260701693, −6.20784997886688426097816444136, −5.59059088843827877611302350792, −5.55710464726469030237342899036, −5.51410301829354224537251271185, −5.32012189263531333891628838934, −4.97322508897565073991685188899, −4.89133280648231016471115972636, −4.82629302905791470748907162388, −4.59969188440715052594304183628, −4.19831349910687715367917400501, −4.01879701112044688376517947091, −3.92367179937230769354795801908, −3.80601454311644342166713331173, −3.27093477526377368493499518590, −3.08947770339262508772239584255, −2.81598929201090046958090229333, −2.13562660544633195124668524043, −1.88297240543891209871058517236, −1.73727295341316230826782990350, −1.44806255035389705206448920758, −1.34044347014290301584258655770, 0, 0, 0, 0, 1.34044347014290301584258655770, 1.44806255035389705206448920758, 1.73727295341316230826782990350, 1.88297240543891209871058517236, 2.13562660544633195124668524043, 2.81598929201090046958090229333, 3.08947770339262508772239584255, 3.27093477526377368493499518590, 3.80601454311644342166713331173, 3.92367179937230769354795801908, 4.01879701112044688376517947091, 4.19831349910687715367917400501, 4.59969188440715052594304183628, 4.82629302905791470748907162388, 4.89133280648231016471115972636, 4.97322508897565073991685188899, 5.32012189263531333891628838934, 5.51410301829354224537251271185, 5.55710464726469030237342899036, 5.59059088843827877611302350792, 6.20784997886688426097816444136, 6.28275183223450232246260701693, 6.31895140748792489154506104026, 6.78095581366020259871362525636, 6.83485545036000741627496619751

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