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Properties

Label 8-51e8-1.1-c1e4-0-5
Degree $8$
Conductor $4.577\times 10^{13}$
Sign $1$
Analytic cond. $186067.$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 4·5-s − 8·7-s + 4·11-s − 4·13-s + 6·16-s − 12·19-s − 16·20-s + 12·23-s − 2·25-s + 32·28-s + 8·29-s − 8·31-s − 32·35-s − 24·37-s + 12·41-s + 4·43-s − 16·44-s − 8·47-s + 16·49-s + 16·52-s − 8·53-s + 16·55-s − 24·59-s − 24·61-s − 16·65-s + 8·67-s + ⋯
L(s)  = 1  − 2·4-s + 1.78·5-s − 3.02·7-s + 1.20·11-s − 1.10·13-s + 3/2·16-s − 2.75·19-s − 3.57·20-s + 2.50·23-s − 2/5·25-s + 6.04·28-s + 1.48·29-s − 1.43·31-s − 5.40·35-s − 3.94·37-s + 1.87·41-s + 0.609·43-s − 2.41·44-s − 1.16·47-s + 16/7·49-s + 2.21·52-s − 1.09·53-s + 2.15·55-s − 3.12·59-s − 3.07·61-s − 1.98·65-s + 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 17^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 17^{8}\)
Sign: $1$
Analytic conductor: \(186067.\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 17^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
17 \( 1 \)
good2$C_2^2:C_4$ \( 1 + p^{2} T^{2} + 5 p T^{4} + p^{4} T^{6} + p^{4} T^{8} \)
5$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 18 T^{2} - 48 T^{3} + 131 T^{4} - 48 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 + 8 T + 48 T^{2} + 184 T^{3} + 576 T^{4} + 184 p T^{5} + 48 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
11$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 38 T^{2} - 128 T^{3} + 595 T^{4} - 128 p T^{5} + 38 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 38 T^{2} + 88 T^{3} + 603 T^{4} + 88 p T^{5} + 38 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 12 T + 94 T^{2} + 512 T^{3} + 2483 T^{4} + 512 p T^{5} + 94 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3547 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - 4 T + 44 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 76 T^{2} + 552 T^{3} + 3422 T^{4} + 552 p T^{5} + 76 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 344 T^{2} + 3304 T^{3} + 23424 T^{4} + 3304 p T^{5} + 344 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 - 12 T + 162 T^{2} - 1472 T^{3} + 9923 T^{4} - 1472 p T^{5} + 162 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 - 4 T + 94 T^{2} - 320 T^{3} + 5683 T^{4} - 320 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 116 T^{2} + 520 T^{3} + 122 p T^{4} + 520 p T^{5} + 116 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 156 T^{2} + 1112 T^{3} + 11414 T^{4} + 1112 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 432 T^{2} + 4856 T^{3} + 44528 T^{4} + 4856 p T^{5} + 432 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 376 T^{2} + 4280 T^{3} + 37376 T^{4} + 4280 p T^{5} + 376 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 272 T^{2} - 1592 T^{3} + 27474 T^{4} - 1592 p T^{5} + 272 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 448 T^{2} + 5448 T^{3} + 53602 T^{4} + 5448 p T^{5} + 448 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 44 T^{2} - 200 T^{3} - 3482 T^{4} - 200 p T^{5} + 44 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 264 T^{2} - 144 T^{3} + 29168 T^{4} - 144 p T^{5} + 264 p^{2} T^{6} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 8 T + 212 T^{2} + 1576 T^{3} + 21142 T^{4} + 1576 p T^{5} + 212 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 16 T + 336 T^{2} + 3376 T^{3} + 41408 T^{4} + 3376 p T^{5} + 336 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 16 T + 428 T^{2} - 4432 T^{3} + 63582 T^{4} - 4432 p T^{5} + 428 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.58354101543640097092219953324, −6.51400071271178616885220632534, −6.25067105840251398932134681411, −6.19136030315264821898765592244, −6.04526707583938296967807065928, −5.62731548425327296165366908392, −5.49498568374749409846141861377, −5.29522271460068380067330445346, −5.11771040661659034477370238311, −4.67420024312016488704910854298, −4.57759223355770108440913130420, −4.47209753354850458869775024937, −4.42195649293209060575086107868, −3.85048353659913843766127411185, −3.76110041973037481178882475275, −3.56931942484959236934649365329, −3.25892384349533540897592667229, −3.03730836147700757946503078364, −2.79027032749175045810635394725, −2.73103065078823272921756603439, −2.34091925276441276809622452990, −1.85889161703114720341641743882, −1.66233840221327300046064212195, −1.40631181104336934884506639715, −1.21205278916614697690086478316, 0, 0, 0, 0, 1.21205278916614697690086478316, 1.40631181104336934884506639715, 1.66233840221327300046064212195, 1.85889161703114720341641743882, 2.34091925276441276809622452990, 2.73103065078823272921756603439, 2.79027032749175045810635394725, 3.03730836147700757946503078364, 3.25892384349533540897592667229, 3.56931942484959236934649365329, 3.76110041973037481178882475275, 3.85048353659913843766127411185, 4.42195649293209060575086107868, 4.47209753354850458869775024937, 4.57759223355770108440913130420, 4.67420024312016488704910854298, 5.11771040661659034477370238311, 5.29522271460068380067330445346, 5.49498568374749409846141861377, 5.62731548425327296165366908392, 6.04526707583938296967807065928, 6.19136030315264821898765592244, 6.25067105840251398932134681411, 6.51400071271178616885220632534, 6.58354101543640097092219953324

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