Properties

Label 8-2736e4-1.1-c1e4-0-5
Degree $8$
Conductor $5.604\times 10^{13}$
Sign $1$
Analytic cond. $227810.$
Root an. cond. $4.67408$
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  + 4·7-s + 6·13-s + 12·17-s − 16·19-s − 8·25-s − 2·43-s + 12·47-s − 6·49-s − 24·59-s + 2·61-s + 6·67-s − 12·71-s + 14·73-s − 18·79-s + 24·91-s − 36·97-s − 36·101-s − 24·107-s + 48·119-s + 4·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.66·13-s + 2.91·17-s − 3.67·19-s − 8/5·25-s − 0.304·43-s + 1.75·47-s − 6/7·49-s − 3.12·59-s + 0.256·61-s + 0.733·67-s − 1.42·71-s + 1.63·73-s − 2.02·79-s + 2.51·91-s − 3.65·97-s − 3.58·101-s − 2.32·107-s + 4.40·119-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{8} \cdot 19^{4}\)
Sign: $1$
Analytic conductor: \(227810.\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2736} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{8} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.221399333\)
\(L(\frac12)\) \(\approx\) \(1.221399333\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$D_{4}$ \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \)
13$D_4\times C_2$ \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 12 T + 92 T^{2} - 528 T^{3} + 2463 T^{4} - 528 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2^3$ \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^3$ \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \)
37$D_4\times C_2$ \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 + 2 T - 59 T^{2} - 46 T^{3} + 1948 T^{4} - 46 p T^{5} - 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 12 T + 152 T^{2} - 1248 T^{3} + 10863 T^{4} - 1248 p T^{5} + 152 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^3$ \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \)
59$D_4\times C_2$ \( 1 + 24 T + 320 T^{2} + 3312 T^{3} + 28071 T^{4} + 3312 p T^{5} + 320 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
61$D_4\times C_2$ \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$D_4\times C_2$ \( 1 - 6 T + 77 T^{2} - 390 T^{3} + 540 T^{4} - 390 p T^{5} + 77 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
79$D_4\times C_2$ \( 1 + 18 T + 275 T^{2} + 3006 T^{3} + 30180 T^{4} + 3006 p T^{5} + 275 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
83$D_4\times C_2$ \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^3$ \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.10763195428405817302696192878, −6.10155366383122081620147513479, −5.85091007390287389818891282118, −5.56030380873662557644912392553, −5.55030930933979473501242946355, −5.36880355583514362488468680838, −5.00745894211047923975235024604, −4.81226828671111579549847094280, −4.53292551631563354012396507438, −4.32512054519750709689735791869, −4.08562092367095796951506087054, −3.98931447689414135389275425362, −3.80260676877354525008802544650, −3.71469928791547836973468570659, −3.20165926242682729976985538956, −2.85211178104411688952897683970, −2.84718886001384022433994239774, −2.64571557677682323745443393841, −1.99781584331197555497295916155, −1.96230387895430539692664771612, −1.46362198568462119419152114756, −1.44449615105317904145381037266, −1.42424820199624414917069757087, −0.74620476490508314624978228702, −0.16438642888878608902395100059, 0.16438642888878608902395100059, 0.74620476490508314624978228702, 1.42424820199624414917069757087, 1.44449615105317904145381037266, 1.46362198568462119419152114756, 1.96230387895430539692664771612, 1.99781584331197555497295916155, 2.64571557677682323745443393841, 2.84718886001384022433994239774, 2.85211178104411688952897683970, 3.20165926242682729976985538956, 3.71469928791547836973468570659, 3.80260676877354525008802544650, 3.98931447689414135389275425362, 4.08562092367095796951506087054, 4.32512054519750709689735791869, 4.53292551631563354012396507438, 4.81226828671111579549847094280, 5.00745894211047923975235024604, 5.36880355583514362488468680838, 5.55030930933979473501242946355, 5.56030380873662557644912392553, 5.85091007390287389818891282118, 6.10155366383122081620147513479, 6.10763195428405817302696192878

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