Properties

Label 8-1815e4-1.1-c1e4-0-1
Degree $8$
Conductor $1.085\times 10^{13}$
Sign $1$
Analytic cond. $44117.9$
Root an. cond. $3.80694$
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  − 4·3-s + 4·5-s + 10·9-s − 16·15-s + 5·16-s − 8·23-s + 10·25-s − 20·27-s + 16·31-s + 40·45-s − 20·48-s − 8·49-s + 16·53-s + 8·59-s − 24·67-s + 32·69-s − 8·71-s − 40·75-s + 20·80-s + 35·81-s + 24·89-s − 64·93-s + 48·97-s − 24·103-s + 16·113-s − 32·115-s + 20·125-s + ⋯
L(s)  = 1  − 2.30·3-s + 1.78·5-s + 10/3·9-s − 4.13·15-s + 5/4·16-s − 1.66·23-s + 2·25-s − 3.84·27-s + 2.87·31-s + 5.96·45-s − 2.88·48-s − 8/7·49-s + 2.19·53-s + 1.04·59-s − 2.93·67-s + 3.85·69-s − 0.949·71-s − 4.61·75-s + 2.23·80-s + 35/9·81-s + 2.54·89-s − 6.63·93-s + 4.87·97-s − 2.36·103-s + 1.50·113-s − 2.98·115-s + 1.78·125-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(44117.9\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.341639587\)
\(L(\frac12)\) \(\approx\) \(3.341639587\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$D_4\times C_2$ \( 1 - 5 T^{4} + p^{4} T^{8} \)
7$C_2^2 \wr C_2$ \( 1 + 8 T^{2} + 62 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
13$C_2^2 \wr C_2$ \( 1 + 8 T^{2} + 302 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 + 48 T^{2} + 1102 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 + 44 T^{2} + 998 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 - 12 T^{2} + 1510 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + 68 T^{2} + 2646 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2 \wr C_2$ \( 1 + 40 T^{2} + 3630 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2$ \( ( 1 + p T^{2} )^{4} \)
53$D_{4}$ \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \)
59$D_{4}$ \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2 \wr C_2$ \( 1 + 164 T^{2} + 13334 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 + 232 T^{2} + 23646 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2 \wr C_2$ \( 1 + 284 T^{2} + 32438 T^{4} + 284 p^{2} T^{6} + p^{4} T^{8} \)
83$C_2^2 \wr C_2$ \( 1 + 144 T^{2} + 10174 T^{4} + 144 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
97$D_{4}$ \( ( 1 - 24 T + 286 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
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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

−6.50130593037571381401084912465, −6.25392072243034789240084593534, −6.08330859910613361759280295000, −5.95339495665438201848916260540, −5.77856292238863530929805054913, −5.73765480492050181214723266192, −5.28754219451573072544295766581, −5.22050649993004836452360529245, −5.05939950189234409119949628967, −4.56894558768323315605813803454, −4.50450983148574077854431798202, −4.47847650461691872698979398435, −4.18807683843940616296336582716, −3.60324089075882518100245231495, −3.54141246293486544080317345458, −3.35905797732878013538847531139, −2.79906935984336969063010067633, −2.55365974040998094112402116431, −2.47519976775313353024965436545, −1.84132316347881496148490324196, −1.71896941413212670191744613550, −1.58103459079575489535843938290, −1.01733217097911136239513203341, −0.65035874800630818736178014695, −0.56823620674354587420955119998, 0.56823620674354587420955119998, 0.65035874800630818736178014695, 1.01733217097911136239513203341, 1.58103459079575489535843938290, 1.71896941413212670191744613550, 1.84132316347881496148490324196, 2.47519976775313353024965436545, 2.55365974040998094112402116431, 2.79906935984336969063010067633, 3.35905797732878013538847531139, 3.54141246293486544080317345458, 3.60324089075882518100245231495, 4.18807683843940616296336582716, 4.47847650461691872698979398435, 4.50450983148574077854431798202, 4.56894558768323315605813803454, 5.05939950189234409119949628967, 5.22050649993004836452360529245, 5.28754219451573072544295766581, 5.73765480492050181214723266192, 5.77856292238863530929805054913, 5.95339495665438201848916260540, 6.08330859910613361759280295000, 6.25392072243034789240084593534, 6.50130593037571381401084912465

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