Properties

Label 8-2475e4-1.1-c1e4-0-3
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $152548.$
Root an. cond. $4.44555$
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-s − 4·11-s − 3·16-s − 12·19-s − 4·29-s − 6·31-s − 6·41-s + 4·44-s + 26·49-s − 34·59-s − 22·61-s + 3·64-s − 6·71-s + 12·76-s + 38·79-s + 52·89-s + 14·101-s − 10·109-s + 4·116-s + 10·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.20·11-s − 3/4·16-s − 2.75·19-s − 0.742·29-s − 1.07·31-s − 0.937·41-s + 0.603·44-s + 26/7·49-s − 4.42·59-s − 2.81·61-s + 3/8·64-s − 0.712·71-s + 1.37·76-s + 4.27·79-s + 5.51·89-s + 1.39·101-s − 0.957·109-s + 0.371·116-s + 0.909·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(152548.\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2475} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.7657131914\)
\(L(\frac12)\) \(\approx\) \(0.7657131914\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 + T )^{4} \)
good2$D_4\times C_2$ \( 1 + T^{2} + p^{2} T^{4} + p^{2} T^{6} + p^{4} T^{8} \)
7$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 31 T^{2} + 472 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 35 T^{2} + 676 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{4} \)
23$D_4\times C_2$ \( 1 - 43 T^{2} + 1176 T^{4} - 43 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 65 T^{2} + 3688 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 91 T^{2} + 5424 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8} \)
47$C_4\times C_2$ \( 1 - 67 T^{2} + 4584 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 - 160 T^{2} + 11406 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 17 T + 186 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 11 T + 148 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 191 T^{2} + 18060 T^{4} - 191 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 + 3 T + 38 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 28 T^{2} - 6554 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 19 T + 210 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{4} \)
89$C_4$ \( ( 1 - 26 T + 330 T^{2} - 26 p T^{3} + p^{2} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 346 T^{2} + 48475 T^{4} - 346 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.48169996566347366923863943437, −6.17025613538500752898514655422, −6.06148699124727451510274318033, −5.64304869033236795391955112728, −5.35165299846590276613134626846, −5.33160750654894310885163737557, −5.22407774327712962172762625584, −4.67080445610505037692380428754, −4.64979561804230755030718427824, −4.57054227379150396305312024395, −4.27337587773533123366520800000, −4.07067266375418368450761482522, −3.73982798649650846271020334769, −3.66508350927003263265020815331, −3.11483625011627001251427772206, −3.06817992036493567550223452121, −3.06313082392782616958415860843, −2.20384145914935899642137216493, −2.16484987816460293698095045957, −2.14094245115857298600516448812, −2.06576001030199132167062658117, −1.39513690258621475943401545764, −1.09546783326443414423385665230, −0.39716261025333494909450183199, −0.26658342085501437107866700878, 0.26658342085501437107866700878, 0.39716261025333494909450183199, 1.09546783326443414423385665230, 1.39513690258621475943401545764, 2.06576001030199132167062658117, 2.14094245115857298600516448812, 2.16484987816460293698095045957, 2.20384145914935899642137216493, 3.06313082392782616958415860843, 3.06817992036493567550223452121, 3.11483625011627001251427772206, 3.66508350927003263265020815331, 3.73982798649650846271020334769, 4.07067266375418368450761482522, 4.27337587773533123366520800000, 4.57054227379150396305312024395, 4.64979561804230755030718427824, 4.67080445610505037692380428754, 5.22407774327712962172762625584, 5.33160750654894310885163737557, 5.35165299846590276613134626846, 5.64304869033236795391955112728, 6.06148699124727451510274318033, 6.17025613538500752898514655422, 6.48169996566347366923863943437

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