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Properties

Label 8-136e4-1.1-c0e4-0-0
Degree $8$
Conductor $342102016$
Sign $1$
Analytic cond. $2.12218\times 10^{-5}$
Root an. cond. $0.260524$
Motivic weight $0$
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  − 2·9-s − 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s − 4·97-s + 8·99-s + 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2·9-s − 4·11-s − 16-s + 4·43-s − 4·59-s + 2·81-s + 4·83-s − 4·97-s + 8·99-s + 4·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 4·176-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(2.12218\times 10^{-5}\)
Root analytic conductor: \(0.260524\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 17^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1260520096\)
\(L(\frac12)\) \(\approx\) \(0.1260520096\)
\(L(1)\) 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$C_2^2$ \( 1 + T^{4} \)
17$C_2^2$ \( 1 + T^{4} \)
good3$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
5$C_4\times C_2$ \( 1 + T^{8} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
13$C_2$ \( ( 1 + T^{2} )^{4} \)
19$C_2^2$ \( ( 1 + T^{4} )^{2} \)
23$C_4\times C_2$ \( 1 + T^{8} \)
29$C_4\times C_2$ \( 1 + T^{8} \)
31$C_4\times C_2$ \( 1 + T^{8} \)
37$C_4\times C_2$ \( 1 + T^{8} \)
41$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T^{2} )^{4} \)
53$C_2^2$ \( ( 1 + T^{4} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \)
61$C_4\times C_2$ \( 1 + T^{8} \)
67$C_2^2$ \( ( 1 + T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2$$\times$$C_2^2$ \( ( 1 + T^{2} )^{2}( 1 + T^{4} ) \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_2$ \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \)
89$C_2^2$ \( ( 1 + T^{4} )^{2} \)
97$C_1$$\times$$C_2^2$ \( ( 1 + T )^{4}( 1 + T^{4} ) \)
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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

−10.05592689375429114078977474895, −9.718940012970763598304128826386, −9.483287034266436882843707780577, −9.069770138442056382190040664788, −8.779608712820201062561950996831, −8.776340600704764084919352485455, −8.175611521551417626168374837597, −8.107939694813905132159077763354, −7.58492444758015215348649087670, −7.58372770452251988614410918593, −7.52382658591326990188196679549, −6.94952358543983213480756466490, −6.30848864215535830980669951229, −6.03618322896740243925109865513, −5.96652199153184866586418734347, −5.50596068933523817240724901115, −5.33526846793379661168496317593, −4.80109863993334326261410933505, −4.76271806022856407544235163714, −4.33544363119264524956986759978, −3.48611060907769416643412290216, −3.21670837177715010608350948468, −2.66511094688412949363541073762, −2.45018497680460612401004236085, −2.29087843910759297804616113393, 2.29087843910759297804616113393, 2.45018497680460612401004236085, 2.66511094688412949363541073762, 3.21670837177715010608350948468, 3.48611060907769416643412290216, 4.33544363119264524956986759978, 4.76271806022856407544235163714, 4.80109863993334326261410933505, 5.33526846793379661168496317593, 5.50596068933523817240724901115, 5.96652199153184866586418734347, 6.03618322896740243925109865513, 6.30848864215535830980669951229, 6.94952358543983213480756466490, 7.52382658591326990188196679549, 7.58372770452251988614410918593, 7.58492444758015215348649087670, 8.107939694813905132159077763354, 8.175611521551417626168374837597, 8.776340600704764084919352485455, 8.779608712820201062561950996831, 9.069770138442056382190040664788, 9.483287034266436882843707780577, 9.718940012970763598304128826386, 10.05592689375429114078977474895

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