Properties

Label 8-1452e4-1.1-c3e4-0-1
Degree $8$
Conductor $4.445\times 10^{12}$
Sign $1$
Analytic cond. $5.38679\times 10^{7}$
Root an. cond. $9.25585$
Motivic weight $3$
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  + 12·3-s + 12·5-s + 90·9-s + 144·15-s + 156·23-s − 56·25-s + 540·27-s + 184·31-s − 176·37-s + 1.08e3·45-s + 852·47-s + 104·49-s + 924·53-s − 360·59-s − 176·67-s + 1.87e3·69-s + 3.27e3·71-s − 672·75-s + 2.83e3·81-s − 3.14e3·89-s + 2.20e3·93-s + 2.76e3·97-s − 2.11e3·111-s + 5.68e3·113-s + 1.87e3·115-s − 588·125-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.07·5-s + 10/3·9-s + 2.47·15-s + 1.41·23-s − 0.447·25-s + 3.84·27-s + 1.06·31-s − 0.782·37-s + 3.57·45-s + 2.64·47-s + 0.303·49-s + 2.39·53-s − 0.794·59-s − 0.320·67-s + 3.26·69-s + 5.47·71-s − 1.03·75-s + 35/9·81-s − 3.74·89-s + 2.46·93-s + 2.89·97-s − 1.80·111-s + 4.73·113-s + 1.51·115-s − 0.420·125-s + 0.000698·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(5.38679\times 10^{7}\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{4} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(45.79784176\)
\(L(\frac12)\) \(\approx\) \(45.79784176\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - p T )^{4} \)
11 \( 1 \)
good5$D_{4}$ \( ( 1 - 6 T + 82 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
7$C_2^2 \wr C_2$ \( 1 - 104 T^{2} + 78702 T^{4} - 104 p^{6} T^{6} + p^{12} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 - 2768 T^{2} + 2845806 T^{4} - 2768 p^{6} T^{6} + p^{12} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 + 6476 T^{2} + 57842214 T^{4} + 6476 p^{6} T^{6} + p^{12} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 + 2164 T^{2} - 82074 p^{2} T^{4} + 2164 p^{6} T^{6} + p^{12} T^{8} \)
23$D_{4}$ \( ( 1 - 78 T + 4438 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
29$C_2^2 \wr C_2$ \( 1 + 49820 T^{2} + 1699129014 T^{4} + 49820 p^{6} T^{6} + p^{12} T^{8} \)
31$D_{4}$ \( ( 1 - 92 T + 4350 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 88 T + 96870 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + 56012 T^{2} + 2348498886 T^{4} + 56012 p^{6} T^{6} + p^{12} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 + 129460 T^{2} + 16787738166 T^{4} + 129460 p^{6} T^{6} + p^{12} T^{8} \)
47$D_{4}$ \( ( 1 - 426 T + 174958 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
53$D_{4}$ \( ( 1 - 462 T + 311290 T^{2} - 462 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 + 180 T + 333190 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 + 750208 T^{2} + 243740388750 T^{4} + 750208 p^{6} T^{6} + p^{12} T^{8} \)
67$D_{4}$ \( ( 1 + 88 T + 501510 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 - 1638 T + 1356670 T^{2} - 1638 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 + 1504516 T^{2} + 867971676390 T^{4} + 1504516 p^{6} T^{6} + p^{12} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 + 842152 T^{2} + 443696062830 T^{4} + 842152 p^{6} T^{6} + p^{12} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 - 7204 T^{2} + 653870781942 T^{4} - 7204 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 1572 T + 1956934 T^{2} + 1572 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 1384 T + 2144910 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.70577393734156938622596511373, −5.95702686102702255678771948414, −5.94135877940455269870778911556, −5.93658975796975407551581997332, −5.58990501830212038642038071208, −5.20772715022042992304312754047, −5.10769640926937496049065384384, −4.77740081798555295940378887349, −4.64469852346036868989481359335, −4.20207611072484682123459720503, −4.07887964698170825706428560282, −3.92246033015887628218326799408, −3.66796593364961174381741754479, −3.17482008660312470183075688649, −3.08673952237412565869921676212, −3.07255390273636188465303941071, −2.64997696928835569545522501455, −2.19381495135116629200561602120, −2.07834165726115336005130340556, −2.01564360986475615084385196481, −1.89796751765664286378818172580, −1.15465518748266874902934504586, −0.867208301481812454103577117427, −0.843433896825307915882105291124, −0.46451468727228732299277996865, 0.46451468727228732299277996865, 0.843433896825307915882105291124, 0.867208301481812454103577117427, 1.15465518748266874902934504586, 1.89796751765664286378818172580, 2.01564360986475615084385196481, 2.07834165726115336005130340556, 2.19381495135116629200561602120, 2.64997696928835569545522501455, 3.07255390273636188465303941071, 3.08673952237412565869921676212, 3.17482008660312470183075688649, 3.66796593364961174381741754479, 3.92246033015887628218326799408, 4.07887964698170825706428560282, 4.20207611072484682123459720503, 4.64469852346036868989481359335, 4.77740081798555295940378887349, 5.10769640926937496049065384384, 5.20772715022042992304312754047, 5.58990501830212038642038071208, 5.93658975796975407551581997332, 5.94135877940455269870778911556, 5.95702686102702255678771948414, 6.70577393734156938622596511373

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