Properties

Label 8-1536e4-1.1-c3e4-0-10
Degree $8$
Conductor $5.566\times 10^{12}$
Sign $1$
Analytic cond. $6.74573\times 10^{7}$
Root an. cond. $9.51981$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 12·3-s + 90·9-s − 40·11-s − 120·17-s − 320·25-s + 540·27-s − 480·33-s − 552·41-s − 432·43-s − 832·49-s − 1.44e3·51-s − 1.36e3·59-s − 864·67-s − 240·73-s − 3.84e3·75-s + 2.83e3·81-s − 952·83-s − 1.56e3·89-s − 2.28e3·97-s − 3.60e3·99-s − 944·107-s − 4.20e3·113-s − 4.18e3·121-s − 6.62e3·123-s + 127-s − 5.18e3·129-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s − 1.09·11-s − 1.71·17-s − 2.55·25-s + 3.84·27-s − 2.53·33-s − 2.10·41-s − 1.53·43-s − 2.42·49-s − 3.95·51-s − 3.00·59-s − 1.57·67-s − 0.384·73-s − 5.91·75-s + 35/9·81-s − 1.25·83-s − 1.85·89-s − 2.38·97-s − 3.65·99-s − 0.852·107-s − 3.49·113-s − 3.14·121-s − 4.85·123-s + 0.000698·127-s − 3.53·129-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{36} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(6.74573\times 10^{7}\)
Root analytic conductor: \(9.51981\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{36} \cdot 3^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{4} \)
good5$D_4\times C_2$ \( 1 + 64 p T^{2} + 1986 p^{2} T^{4} + 64 p^{7} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 + 832 T^{2} + 7746 p^{2} T^{4} + 832 p^{6} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 20 T + 2690 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 + 8428 T^{2} + 27382614 T^{4} + 8428 p^{6} T^{6} + p^{12} T^{8} \)
17$D_{4}$ \( ( 1 + 60 T + 10334 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
19$C_2^2$ \( ( 1 + 12750 T^{2} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 7628 T^{2} - 37861626 T^{4} + 7628 p^{6} T^{6} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 21056 T^{2} + 1079985426 T^{4} + 21056 p^{6} T^{6} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 + 75424 T^{2} + 3008245506 T^{4} + 75424 p^{6} T^{6} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 106492 T^{2} + 7210762134 T^{4} + 106492 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 276 T + 155086 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 216 T + 146478 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 132332 T^{2} + 9563109414 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 560768 T^{2} + 122848689714 T^{4} + 560768 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 680 T + 526070 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 345244 T^{2} + 119910783606 T^{4} + 345244 p^{6} T^{6} + p^{12} T^{8} \)
67$D_{4}$ \( ( 1 + 432 T + 416982 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 1199084 T^{2} + 615208348806 T^{4} + 1199084 p^{6} T^{6} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 + 120 T + 707906 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 1369696 T^{2} + 864453694146 T^{4} + 1369696 p^{6} T^{6} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 + 476 T + 1155218 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 780 T + 1357238 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 1140 T + 2147654 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.88076864187030466256985983582, −6.51609326640480561745353792290, −6.44128998051250685127705341066, −6.21905279632793005240688559935, −6.19430345558652033127649799442, −5.49511109386720359164372024274, −5.37787091495853732332074130301, −5.32371925869689798462246958153, −5.11708349830427572788607573639, −4.49059041340007990954171567411, −4.40697523297324033496142007851, −4.38185286182576214168712445325, −4.31426973654103408089355108814, −3.67743230845008799913238868691, −3.52716611123759888175987561333, −3.32173941682719244357383633908, −3.30815850508713374058001871441, −2.70541137801708262113865156073, −2.62964016922666933137456693574, −2.43061232492013199318325766068, −2.28384009029328334797506691889, −1.63259823996904915562747471578, −1.61220443836920412881849349673, −1.40011985458388851642270099713, −1.33046816970863279326688000088, 0, 0, 0, 0, 1.33046816970863279326688000088, 1.40011985458388851642270099713, 1.61220443836920412881849349673, 1.63259823996904915562747471578, 2.28384009029328334797506691889, 2.43061232492013199318325766068, 2.62964016922666933137456693574, 2.70541137801708262113865156073, 3.30815850508713374058001871441, 3.32173941682719244357383633908, 3.52716611123759888175987561333, 3.67743230845008799913238868691, 4.31426973654103408089355108814, 4.38185286182576214168712445325, 4.40697523297324033496142007851, 4.49059041340007990954171567411, 5.11708349830427572788607573639, 5.32371925869689798462246958153, 5.37787091495853732332074130301, 5.49511109386720359164372024274, 6.19430345558652033127649799442, 6.21905279632793005240688559935, 6.44128998051250685127705341066, 6.51609326640480561745353792290, 6.88076864187030466256985983582

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