Properties

Label 8-9702e4-1.1-c1e4-0-1
Degree $8$
Conductor $8.860\times 10^{15}$
Sign $1$
Analytic cond. $3.60208\times 10^{7}$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 10·4-s − 4·5-s − 20·8-s + 16·10-s − 4·11-s + 8·13-s + 35·16-s − 4·17-s + 12·19-s − 40·20-s + 16·22-s + 8·23-s − 32·26-s + 8·29-s + 4·31-s − 56·32-s + 16·34-s + 8·37-s − 48·38-s + 80·40-s − 12·41-s − 8·43-s − 40·44-s − 32·46-s − 4·47-s + 80·52-s + ⋯
L(s)  = 1  − 2.82·2-s + 5·4-s − 1.78·5-s − 7.07·8-s + 5.05·10-s − 1.20·11-s + 2.21·13-s + 35/4·16-s − 0.970·17-s + 2.75·19-s − 8.94·20-s + 3.41·22-s + 1.66·23-s − 6.27·26-s + 1.48·29-s + 0.718·31-s − 9.89·32-s + 2.74·34-s + 1.31·37-s − 7.78·38-s + 12.6·40-s − 1.87·41-s − 1.21·43-s − 6.03·44-s − 4.71·46-s − 0.583·47-s + 11.0·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(3.60208\times 10^{7}\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9702} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7185212773\)
\(L(\frac12)\) \(\approx\) \(0.7185212773\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 48 T^{2} - 232 T^{3} + 882 T^{4} - 232 p T^{5} + 48 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 16 T^{2} + 92 T^{3} + 742 T^{4} + 92 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 116 T^{2} - 708 T^{3} + 3658 T^{4} - 708 p T^{5} + 116 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 4 p T^{2} - 520 T^{3} + 3190 T^{4} - 520 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 84 T^{2} - 632 T^{3} + 3254 T^{4} - 632 p T^{5} + 84 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 36 T^{2} - 44 T^{3} + 1274 T^{4} - 44 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 116 T^{2} - 792 T^{3} + 5862 T^{4} - 792 p T^{5} + 116 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 80 T^{2} - 76 T^{3} - 1786 T^{4} - 76 p T^{5} + 80 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 140 T^{2} + 936 T^{3} + 8358 T^{4} + 936 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 164 T^{2} + 572 T^{3} + 11002 T^{4} + 572 p T^{5} + 164 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 52 T^{2} - 280 T^{3} + 3750 T^{4} - 280 p T^{5} + 52 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 92 T^{2} - 512 T^{3} + 3446 T^{4} - 512 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 24 T + 336 T^{2} - 3768 T^{3} + 33714 T^{4} - 3768 p T^{5} + 336 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 228 T^{2} + 1576 T^{3} + 21606 T^{4} + 1576 p T^{5} + 228 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 60 T^{2} + 744 T^{3} + 10790 T^{4} + 744 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 224 T^{2} - 796 T^{3} + 22758 T^{4} - 796 p T^{5} + 224 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 180 T^{2} + 904 T^{3} + 15990 T^{4} + 904 p T^{5} + 180 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 228 T^{2} + 860 T^{3} + 26058 T^{4} + 860 p T^{5} + 228 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 480 T^{2} + 6424 T^{3} + 69314 T^{4} + 6424 p T^{5} + 480 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 128 T^{2} + 1280 T^{3} + 4866 T^{4} + 1280 p T^{5} + 128 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.50361048404764323280807173334, −5.11771542428304673747607150832, −5.06207505048478503417029193598, −4.95166239804956323136239028614, −4.80905643267898948794957069397, −4.30128557941788803465807220847, −4.21995658011911024667627159144, −3.97705405554895215965815780523, −3.82658408903865063364265524386, −3.57831477886499631822968295562, −3.36553942625143503710044523424, −3.33641082057976988580846083004, −3.07274148683137598650688776636, −2.67332531852282836318390317425, −2.57349429624634832332477361136, −2.55373992306927247605057385042, −2.53190155302618950413180578835, −1.67065542604690892796760911574, −1.57100030283287942985611410501, −1.44543148355820861875495028477, −1.40925759376384295005214542228, −0.856578081408337407676950322815, −0.76948084543206307722149353370, −0.46830935610164087318279491479, −0.26329443169943836493114912312, 0.26329443169943836493114912312, 0.46830935610164087318279491479, 0.76948084543206307722149353370, 0.856578081408337407676950322815, 1.40925759376384295005214542228, 1.44543148355820861875495028477, 1.57100030283287942985611410501, 1.67065542604690892796760911574, 2.53190155302618950413180578835, 2.55373992306927247605057385042, 2.57349429624634832332477361136, 2.67332531852282836318390317425, 3.07274148683137598650688776636, 3.33641082057976988580846083004, 3.36553942625143503710044523424, 3.57831477886499631822968295562, 3.82658408903865063364265524386, 3.97705405554895215965815780523, 4.21995658011911024667627159144, 4.30128557941788803465807220847, 4.80905643267898948794957069397, 4.95166239804956323136239028614, 5.06207505048478503417029193598, 5.11771542428304673747607150832, 5.50361048404764323280807173334

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