Properties

Label 8-1900e4-1.1-c1e4-0-0
Degree $8$
Conductor $1.303\times 10^{13}$
Sign $1$
Analytic cond. $52981.3$
Root an. cond. $3.89507$
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  − 2·9-s − 12·11-s − 14·19-s + 2·29-s − 20·31-s − 4·41-s − 4·49-s − 2·59-s + 14·61-s − 30·71-s − 2·79-s + 9·81-s + 34·89-s + 24·99-s − 22·101-s + 14·109-s + 46·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯
L(s)  = 1  − 2/3·9-s − 3.61·11-s − 3.21·19-s + 0.371·29-s − 3.59·31-s − 0.624·41-s − 4/7·49-s − 0.260·59-s + 1.79·61-s − 3.56·71-s − 0.225·79-s + 81-s + 3.60·89-s + 2.41·99-s − 2.18·101-s + 1.34·109-s + 4.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(52981.3\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2052700009\)
\(L(\frac12)\) \(\approx\) \(0.2052700009\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
good3$C_2^3$ \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
13$C_2^2$$\times$$C_2^2$ \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
23$C_2^3$ \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^3$ \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^3$ \( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 15 T + 154 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 2 T^{2} - 5325 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 17 T + 200 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 50 T^{2} - 6909 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.53778704800134409740544086563, −6.33037633272993282165111450303, −6.10714371964561215871727989540, −5.77153856994110900655094108920, −5.56379409686943330221828030992, −5.52780305382413688666017398860, −5.51834184769869826589510585644, −4.92898894419454928772284863214, −4.91096878541930056402387134607, −4.77308700900269260062626641040, −4.37702454940207645751945942305, −4.12841700320055809496326994545, −4.11532198377796116124187267334, −3.42111802997040223553139115874, −3.39158629347785118188882476289, −3.32294012882191153578081542509, −2.92295046690702591822291711862, −2.47654431552578436207129508419, −2.38117468400252566111854659720, −2.12914862500964996315940573592, −2.07558944369751907460501090615, −1.67857715275242319453313053644, −1.15936733279726361671232939108, −0.31738247707135691938915555307, −0.18587966166460704873263648031, 0.18587966166460704873263648031, 0.31738247707135691938915555307, 1.15936733279726361671232939108, 1.67857715275242319453313053644, 2.07558944369751907460501090615, 2.12914862500964996315940573592, 2.38117468400252566111854659720, 2.47654431552578436207129508419, 2.92295046690702591822291711862, 3.32294012882191153578081542509, 3.39158629347785118188882476289, 3.42111802997040223553139115874, 4.11532198377796116124187267334, 4.12841700320055809496326994545, 4.37702454940207645751945942305, 4.77308700900269260062626641040, 4.91096878541930056402387134607, 4.92898894419454928772284863214, 5.51834184769869826589510585644, 5.52780305382413688666017398860, 5.56379409686943330221828030992, 5.77153856994110900655094108920, 6.10714371964561215871727989540, 6.33037633272993282165111450303, 6.53778704800134409740544086563

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