Properties

Label 8-3072e4-1.1-c1e4-0-12
Degree $8$
Conductor $8.906\times 10^{13}$
Sign $1$
Analytic cond. $362070.$
Root an. cond. $4.95278$
Motivic weight $1$
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  − 4·3-s − 4·5-s + 4·7-s + 10·9-s − 8·13-s + 16·15-s − 16·21-s − 20·27-s − 12·29-s + 12·31-s − 16·35-s − 16·37-s + 32·39-s − 40·45-s − 4·49-s − 20·53-s − 16·61-s + 40·63-s + 32·65-s + 16·67-s − 8·71-s − 8·73-s + 12·79-s + 35·81-s + 48·87-s − 8·89-s − 32·91-s + ⋯
L(s)  = 1  − 2.30·3-s − 1.78·5-s + 1.51·7-s + 10/3·9-s − 2.21·13-s + 4.13·15-s − 3.49·21-s − 3.84·27-s − 2.22·29-s + 2.15·31-s − 2.70·35-s − 2.63·37-s + 5.12·39-s − 5.96·45-s − 4/7·49-s − 2.74·53-s − 2.04·61-s + 5.03·63-s + 3.96·65-s + 1.95·67-s − 0.949·71-s − 0.936·73-s + 1.35·79-s + 35/9·81-s + 5.14·87-s − 0.847·89-s − 3.35·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{40} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(362070.\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3072} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{40} \cdot 3^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 20 T^{2} - 60 T^{3} + 186 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 20 T^{2} - 32 T^{3} + 230 T^{4} - 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 56 T^{2} + 264 T^{3} + 1122 T^{4} + 264 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 144 T^{2} + 964 T^{3} + 6422 T^{4} + 964 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 200 T^{2} + 1552 T^{3} + 11010 T^{4} + 1552 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 100 T^{2} + 192 T^{3} + 4726 T^{4} + 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 336 T^{2} + 3452 T^{3} + 30134 T^{4} + 3452 p T^{5} + 336 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 296 T^{2} + 2704 T^{3} + 27618 T^{4} + 2704 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 116 T^{2} - 160 T^{3} + 5510 T^{4} - 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 156 T^{2} + 504 T^{3} + 10022 T^{4} + 504 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.59114718406316778683039080392, −6.26028391532533158243871709263, −6.03231731854026104154876893350, −6.00875653261882002845927317322, −5.89542822884915002247658874286, −5.25601866950955791846088824986, −5.23779967898676565664828235354, −5.12144938094795925397581555058, −5.11519260331973458721246701333, −4.74148713213143816435262184885, −4.55227221478712693960031752468, −4.45716802777566801135130492773, −4.40567691563585988635624126163, −3.78337667086529487549485514100, −3.70948368662147059097088814276, −3.68448347064441934312972789376, −3.45552041095801557250812552157, −2.91713018417804372195105863698, −2.68261032884944676707285555881, −2.32203425828271414153679950131, −2.23856332766896901428855623750, −1.64234283423106015247744593109, −1.59695698861670199209041097030, −1.20492633795042585721050146197, −1.14943319673942930481233004926, 0, 0, 0, 0, 1.14943319673942930481233004926, 1.20492633795042585721050146197, 1.59695698861670199209041097030, 1.64234283423106015247744593109, 2.23856332766896901428855623750, 2.32203425828271414153679950131, 2.68261032884944676707285555881, 2.91713018417804372195105863698, 3.45552041095801557250812552157, 3.68448347064441934312972789376, 3.70948368662147059097088814276, 3.78337667086529487549485514100, 4.40567691563585988635624126163, 4.45716802777566801135130492773, 4.55227221478712693960031752468, 4.74148713213143816435262184885, 5.11519260331973458721246701333, 5.12144938094795925397581555058, 5.23779967898676565664828235354, 5.25601866950955791846088824986, 5.89542822884915002247658874286, 6.00875653261882002845927317322, 6.03231731854026104154876893350, 6.26028391532533158243871709263, 6.59114718406316778683039080392

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