Properties

Degree 8
Conductor $ 2^{8} \cdot 3^{12} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 52·4-s − 664·13-s + 1.68e3·16-s − 4.26e3·25-s + 6.13e4·37-s + 1.69e4·49-s + 3.45e4·52-s − 2.12e5·61-s − 3.41e4·64-s − 1.22e5·73-s − 5.48e4·97-s + 2.21e5·100-s − 3.50e5·109-s + 2.91e4·121-s + 127-s + 131-s + 137-s + 139-s − 3.18e6·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.20e6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.62·4-s − 1.08·13-s + 1.64·16-s − 1.36·25-s + 7.36·37-s + 1.00·49-s + 1.77·52-s − 7.32·61-s − 1.04·64-s − 2.70·73-s − 0.592·97-s + 2.21·100-s − 2.82·109-s + 0.180·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 11.9·148-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 3.25·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(8\)
\( N \)  =  \(2^{8} \cdot 3^{12}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{108} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((8,\ 2^{8} \cdot 3^{12} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)
\(L(3)\)  \(\approx\)  \(1.23778\)
\(L(\frac12)\)  \(\approx\)  \(1.23778\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 8. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 7.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2^2$ \( 1 + 13 p^{2} T^{2} + p^{10} T^{4} \)
3 \( 1 \)
good5$C_2^2$ \( ( 1 + 2131 T^{2} + p^{10} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 8471 T^{2} + p^{10} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 14573 T^{2} + p^{10} T^{4} )^{2} \)
13$C_2$ \( ( 1 + 166 T + p^{5} T^{2} )^{4} \)
17$C_2^2$ \( ( 1 - 2151950 T^{2} + p^{10} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 - 4501190 T^{2} + p^{10} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 - 2019266 T^{2} + p^{10} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 - 1012606 p T^{2} + p^{10} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 17147735 T^{2} + p^{10} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 15332 T + p^{5} T^{2} )^{4} \)
41$C_2^2$ \( ( 1 - 129650498 T^{2} + p^{10} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 284996378 T^{2} + p^{10} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 + 408701842 T^{2} + p^{10} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 537758237 T^{2} + p^{10} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 621590410 T^{2} + p^{10} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 53188 T + p^{5} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 1014398666 T^{2} + p^{10} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 2889010114 T^{2} + p^{10} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 30739 T + p^{5} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 1226373986 T^{2} + p^{10} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 7748073619 T^{2} + p^{10} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 6450847934 T^{2} + p^{10} T^{4} )^{2} \)
97$C_2$ \( ( 1 + 13717 T + p^{5} T^{2} )^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.167300936214596725572387606365, −8.968575089485689946828740757456, −8.702323969995413705393106513413, −8.010201427548553396384716944636, −7.967687086925914964170597447543, −7.61570506982922370239171693455, −7.61069747322674671738057090547, −7.35779739041483077347500254128, −6.54084307009892782626923672978, −6.13141339692114794077444533977, −6.09498178415135196849914640881, −5.81299978084502266307040812790, −5.46099797467672802658819417288, −4.72661546642776125940035274986, −4.57721111555965388378629222644, −4.53868569769019495217379944186, −3.97101683524937158435743959909, −3.95371002812051955741869574151, −2.88071851800310368833992851613, −2.80451723785192321413831739066, −2.61647444454060181692975241903, −1.57478998957193279536218672530, −1.33690246075556839003864221758, −0.59143899776432484941606518320, −0.30877576698172027352091276740, 0.30877576698172027352091276740, 0.59143899776432484941606518320, 1.33690246075556839003864221758, 1.57478998957193279536218672530, 2.61647444454060181692975241903, 2.80451723785192321413831739066, 2.88071851800310368833992851613, 3.95371002812051955741869574151, 3.97101683524937158435743959909, 4.53868569769019495217379944186, 4.57721111555965388378629222644, 4.72661546642776125940035274986, 5.46099797467672802658819417288, 5.81299978084502266307040812790, 6.09498178415135196849914640881, 6.13141339692114794077444533977, 6.54084307009892782626923672978, 7.35779739041483077347500254128, 7.61069747322674671738057090547, 7.61570506982922370239171693455, 7.967687086925914964170597447543, 8.010201427548553396384716944636, 8.702323969995413705393106513413, 8.968575089485689946828740757456, 9.167300936214596725572387606365

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