Properties

Label 8-1536e4-1.1-c3e4-0-9
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 − 8·11-s − 344·17-s + 16·19-s − 48·25-s + 540·27-s − 96·33-s − 904·41-s − 832·43-s − 984·49-s − 4.12e3·51-s + 192·57-s − 432·59-s + 416·67-s − 1.05e3·73-s − 576·75-s + 2.83e3·81-s + 1.51e3·83-s + 712·89-s − 2.68e3·97-s − 720·99-s − 6.35e3·107-s + 1.62e3·113-s − 2.46e3·121-s − 1.08e4·123-s + 127-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s − 0.219·11-s − 4.90·17-s + 0.193·19-s − 0.383·25-s + 3.84·27-s − 0.506·33-s − 3.44·41-s − 2.95·43-s − 2.86·49-s − 11.3·51-s + 0.446·57-s − 0.953·59-s + 0.758·67-s − 1.69·73-s − 0.886·75-s + 35/9·81-s + 1.99·83-s + 0.847·89-s − 2.80·97-s − 0.730·99-s − 5.73·107-s + 1.35·113-s − 1.85·121-s − 7.95·123-s + 0.000698·127-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$D_4\times C_2$ \( 1 + 48 T^{2} - 3374 T^{4} + 48 p^{6} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 + 984 T^{2} + 464690 T^{4} + 984 p^{6} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 4 T + 1258 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 + 1802 T^{2} + p^{6} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 86 T + p^{3} T^{2} )^{4} \)
19$D_{4}$ \( ( 1 - 8 T + 8102 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 + 12172 T^{2} + 287491974 T^{4} + 12172 p^{6} T^{6} + p^{12} T^{8} \)
29$D_4\times C_2$ \( 1 + 73872 T^{2} + 2422935538 T^{4} + 73872 p^{6} T^{6} + p^{12} T^{8} \)
31$D_4\times C_2$ \( 1 + 37464 T^{2} + 1927895186 T^{4} + 37464 p^{6} T^{6} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 99940 T^{2} + 6487973718 T^{4} + 99940 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 452 T + 183286 T^{2} + 452 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 416 T + 179750 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 + 150700 T^{2} + 10066811046 T^{4} + 150700 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 + 89904 T^{2} + 34920948754 T^{4} + 89904 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 + 216 T - 33770 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_4\times C_2$ \( 1 + 903812 T^{2} + 307255974326 T^{4} + 903812 p^{6} T^{6} + p^{12} T^{8} \)
67$D_{4}$ \( ( 1 - 208 T + 606710 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 + 1217740 T^{2} + 615534867654 T^{4} + 1217740 p^{6} T^{6} + p^{12} T^{8} \)
73$D_{4}$ \( ( 1 + 528 T - 32270 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 + 1583192 T^{2} + 1093199269970 T^{4} + 1583192 p^{6} T^{6} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 756 T + 1172410 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 - 4 p T + 630614 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 + 1340 T + 1913798 T^{2} + 1340 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
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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

−6.75216452020249590665834601662, −6.58881676036565501107616490093, −6.48327049793507853400148262906, −6.44842511665081320371834946314, −6.26002868990015821007746973236, −5.56214563102896531182711991200, −5.36882830542005170589553233463, −5.04780544919411483998440603516, −5.03802243645960678647981167041, −4.69848356297949465972818941210, −4.49728447311070887115097263529, −4.41618151402258670389041294951, −4.09186810444180981861180227567, −3.68731595490740387335954903876, −3.59812444748920676924109132250, −3.47913373324268398162195739362, −3.17253740980885220498807924839, −2.76923993154289084030544176352, −2.40196362227580942511170118516, −2.39888120271100112736393972015, −2.39763141762723776247304805574, −1.73583016079313234962246232405, −1.55573050187726606072309519603, −1.50875677683888112402350059319, −1.22042340242097089665649742237, 0, 0, 0, 0, 1.22042340242097089665649742237, 1.50875677683888112402350059319, 1.55573050187726606072309519603, 1.73583016079313234962246232405, 2.39763141762723776247304805574, 2.39888120271100112736393972015, 2.40196362227580942511170118516, 2.76923993154289084030544176352, 3.17253740980885220498807924839, 3.47913373324268398162195739362, 3.59812444748920676924109132250, 3.68731595490740387335954903876, 4.09186810444180981861180227567, 4.41618151402258670389041294951, 4.49728447311070887115097263529, 4.69848356297949465972818941210, 5.03802243645960678647981167041, 5.04780544919411483998440603516, 5.36882830542005170589553233463, 5.56214563102896531182711991200, 6.26002868990015821007746973236, 6.44842511665081320371834946314, 6.48327049793507853400148262906, 6.58881676036565501107616490093, 6.75216452020249590665834601662

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