Properties

Label 8-2800e4-1.1-c1e4-0-6
Degree $8$
Conductor $6.147\times 10^{13}$
Sign $1$
Analytic cond. $249885.$
Root an. cond. $4.72843$
Motivic weight $1$
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  + 3·9-s + 8·11-s + 6·19-s − 22·29-s + 20·31-s − 18·41-s − 2·49-s − 4·59-s + 28·61-s + 18·71-s + 2·79-s − 7·81-s + 2·89-s + 24·99-s − 22·109-s + 30·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 18·171-s + ⋯
L(s)  = 1  + 9-s + 2.41·11-s + 1.37·19-s − 4.08·29-s + 3.59·31-s − 2.81·41-s − 2/7·49-s − 0.520·59-s + 3.58·61-s + 2.13·71-s + 0.225·79-s − 7/9·81-s + 0.211·89-s + 2.41·99-s − 2.10·109-s + 2.72·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 1.37·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(249885.\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.433915057\)
\(L(\frac12)\) \(\approx\) \(2.433915057\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7$C_2$ \( ( 1 + T^{2} )^{2} \)
good3$D_4\times C_2$ \( 1 - p T^{2} + 16 T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 - 4 T + 9 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 59 T^{2} + 1444 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 - 3 T + 36 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 59 T^{2} + 1720 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 + 11 T + 84 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 + 69 T^{2} + 2972 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 9 T + 98 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 123 T^{2} + 7136 T^{4} - 123 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \)
53$D_4\times C_2$ \( 1 - 160 T^{2} + 11406 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 + 2 T - 34 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 + 54 T^{2} - 85 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8} \)
71$D_{4}$ \( ( 1 - 9 T + 124 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 139 T^{2} + 14260 T^{4} - 139 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - T + 120 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 143 T^{2} + 10284 T^{4} - 143 p^{2} T^{6} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - T + 72 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
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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

−6.17042539008638506195731407769, −6.13929410612148289682104804336, −5.84390331609604569277444602572, −5.75379142903300602547713940141, −5.23953817822211231278150775528, −5.21555442367481850101466335603, −4.96455660050966067516727673827, −4.93079388262413828629638135593, −4.65110496010331723723461068862, −4.22495129991534024334847668274, −4.02580068298830885842620489189, −3.88331654587880621317949068624, −3.70535988318055846433279057956, −3.66808647126897038840813217162, −3.39631878616851132415780723590, −2.94498433768572369258952784987, −2.92609117601338841310720152516, −2.31501067941884029106351264902, −2.15145876266324209318313177541, −2.06123683314587195319766088841, −1.54546765895884468657678162668, −1.27173748972588923671575047469, −1.11281497066095856328870657308, −1.00244876753282514339032836496, −0.21527709717583578563306915765, 0.21527709717583578563306915765, 1.00244876753282514339032836496, 1.11281497066095856328870657308, 1.27173748972588923671575047469, 1.54546765895884468657678162668, 2.06123683314587195319766088841, 2.15145876266324209318313177541, 2.31501067941884029106351264902, 2.92609117601338841310720152516, 2.94498433768572369258952784987, 3.39631878616851132415780723590, 3.66808647126897038840813217162, 3.70535988318055846433279057956, 3.88331654587880621317949068624, 4.02580068298830885842620489189, 4.22495129991534024334847668274, 4.65110496010331723723461068862, 4.93079388262413828629638135593, 4.96455660050966067516727673827, 5.21555442367481850101466335603, 5.23953817822211231278150775528, 5.75379142903300602547713940141, 5.84390331609604569277444602572, 6.13929410612148289682104804336, 6.17042539008638506195731407769

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