Properties

Label 8-882e4-1.1-c2e4-0-2
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $333591.$
Root an. cond. $4.90232$
Motivic weight $2$
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  − 2·4-s − 6·5-s − 18·11-s − 30·17-s − 6·19-s + 12·20-s − 30·23-s − 5·25-s − 48·29-s + 42·31-s − 62·37-s − 8·43-s + 36·44-s + 174·47-s + 78·53-s + 108·55-s − 78·59-s + 42·61-s + 8·64-s − 58·67-s + 60·68-s + 24·71-s − 318·73-s + 12·76-s + 110·79-s + 180·85-s − 378·89-s + ⋯
L(s)  = 1  − 1/2·4-s − 6/5·5-s − 1.63·11-s − 1.76·17-s − 0.315·19-s + 3/5·20-s − 1.30·23-s − 1/5·25-s − 1.65·29-s + 1.35·31-s − 1.67·37-s − 0.186·43-s + 9/11·44-s + 3.70·47-s + 1.47·53-s + 1.96·55-s − 1.32·59-s + 0.688·61-s + 1/8·64-s − 0.865·67-s + 0.882·68-s + 0.338·71-s − 4.35·73-s + 3/19·76-s + 1.39·79-s + 2.11·85-s − 4.24·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(333591.\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.009359301353\)
\(L(\frac12)\) \(\approx\) \(0.009359301353\)
\(L(2)\) 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$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2$$\times$$C_2^2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2}( 1 + 2 T - 21 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} ) \)
11$D_4\times C_2$ \( 1 + 18 T + 19 T^{2} + 1134 T^{3} + 39180 T^{4} + 1134 p^{2} T^{5} + 19 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \)
13$D_4\times C_2$ \( 1 - 412 T^{2} + 89190 T^{4} - 412 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 + 30 T + 929 T^{2} + 1110 p T^{3} + 1380 p^{2} T^{4} + 1110 p^{3} T^{5} + 929 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 + 6 T + 731 T^{2} + 4314 T^{3} + 390972 T^{4} + 4314 p^{2} T^{5} + 731 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 + 30 T - 221 T^{2} + 1890 T^{3} + 500700 T^{4} + 1890 p^{2} T^{5} - 221 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 24 T + 1754 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 42 T + 1307 T^{2} - 30198 T^{3} + 158508 T^{4} - 30198 p^{2} T^{5} + 1307 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \)
37$D_4\times C_2$ \( 1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 11842 p^{2} T^{5} + 1297 p^{4} T^{6} + 62 p^{6} T^{7} + p^{8} T^{8} \)
41$D_4\times C_2$ \( 1 - 5500 T^{2} + 13185222 T^{4} - 5500 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 4 T + 3630 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 174 T + 17027 T^{2} - 1206690 T^{3} + 65507772 T^{4} - 1206690 p^{2} T^{5} + 17027 p^{4} T^{6} - 174 p^{6} T^{7} + p^{8} T^{8} \)
53$D_4\times C_2$ \( 1 - 78 T - 767 T^{2} - 96174 T^{3} + 21955764 T^{4} - 96174 p^{2} T^{5} - 767 p^{4} T^{6} - 78 p^{6} T^{7} + p^{8} T^{8} \)
59$D_4\times C_2$ \( 1 + 78 T + 5747 T^{2} + 290082 T^{3} + 8773068 T^{4} + 290082 p^{2} T^{5} + 5747 p^{4} T^{6} + 78 p^{6} T^{7} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 42 T + 2033 T^{2} - 60690 T^{3} - 9569868 T^{4} - 60690 p^{2} T^{5} + 2033 p^{4} T^{6} - 42 p^{6} T^{7} + p^{8} T^{8} \)
67$D_4\times C_2$ \( 1 + 58 T - 2405 T^{2} - 186122 T^{3} - 1970756 T^{4} - 186122 p^{2} T^{5} - 2405 p^{4} T^{6} + 58 p^{6} T^{7} + p^{8} T^{8} \)
71$D_{4}$ \( ( 1 - 12 T + 8318 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 318 T + 51257 T^{2} + 5580582 T^{3} + 459199092 T^{4} + 5580582 p^{2} T^{5} + 51257 p^{4} T^{6} + 318 p^{6} T^{7} + p^{8} T^{8} \)
79$D_4\times C_2$ \( 1 - 110 T - 2957 T^{2} - 283250 T^{3} + 112247068 T^{4} - 283250 p^{2} T^{5} - 2957 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 + 380 T^{2} + 89625894 T^{4} + 380 p^{4} T^{6} + p^{8} T^{8} \)
89$D_4\times C_2$ \( 1 + 378 T + 71921 T^{2} + 9182754 T^{3} + 904668996 T^{4} + 9182754 p^{2} T^{5} + 71921 p^{4} T^{6} + 378 p^{6} T^{7} + p^{8} T^{8} \)
97$D_4\times C_2$ \( 1 - 26620 T^{2} + 330657414 T^{4} - 26620 p^{4} T^{6} + p^{8} T^{8} \)
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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

−7.13282959512998953991311057381, −6.97650105561894912647919070514, −6.57455033099203155548181961156, −6.50340051238039404987693781207, −6.14229048973151008689058510889, −5.76910362673495159519430944936, −5.49025276980308452893117260333, −5.47663603788625569465403184449, −5.41898667089045671501414065010, −4.94848447080990388650997581927, −4.46331688877763571588688955919, −4.31649505340004293399770739683, −4.31222031291706094432464058832, −3.91007734863863222852464762648, −3.88876131320758498567255797934, −3.57952353694507419417180154458, −2.99306947250500265075782239028, −2.59883205973950040642520141965, −2.55416207430979637727058779785, −2.52781077489288089435699294788, −1.68309290733084587553488219944, −1.68296562144759337011922319915, −1.09151748166526813216292605729, −0.31361254154237396522775822579, −0.03638163680107825820690484324, 0.03638163680107825820690484324, 0.31361254154237396522775822579, 1.09151748166526813216292605729, 1.68296562144759337011922319915, 1.68309290733084587553488219944, 2.52781077489288089435699294788, 2.55416207430979637727058779785, 2.59883205973950040642520141965, 2.99306947250500265075782239028, 3.57952353694507419417180154458, 3.88876131320758498567255797934, 3.91007734863863222852464762648, 4.31222031291706094432464058832, 4.31649505340004293399770739683, 4.46331688877763571588688955919, 4.94848447080990388650997581927, 5.41898667089045671501414065010, 5.47663603788625569465403184449, 5.49025276980308452893117260333, 5.76910362673495159519430944936, 6.14229048973151008689058510889, 6.50340051238039404987693781207, 6.57455033099203155548181961156, 6.97650105561894912647919070514, 7.13282959512998953991311057381

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