Properties

Label 8-9800e4-1.1-c1e4-0-5
Degree $8$
Conductor $9.224\times 10^{15}$
Sign $1$
Analytic cond. $3.74983\times 10^{7}$
Root an. cond. $8.84609$
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  − 5·9-s + 6·11-s + 4·23-s − 14·29-s + 4·37-s + 16·43-s + 20·53-s + 32·67-s + 4·71-s + 6·79-s + 11·81-s − 30·99-s − 32·107-s + 30·109-s − 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 45·169-s + 173-s + ⋯
L(s)  = 1  − 5/3·9-s + 1.80·11-s + 0.834·23-s − 2.59·29-s + 0.657·37-s + 2.43·43-s + 2.74·53-s + 3.90·67-s + 0.474·71-s + 0.675·79-s + 11/9·81-s − 3.01·99-s − 3.09·107-s + 2.87·109-s − 0.376·113-s − 0.0909·121-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 − 3.46·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(3.74983\times 10^{7}\)
Root analytic conductor: \(8.84609\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.600108301\)
\(L(\frac12)\) \(\approx\) \(8.600108301\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2 \wr C_2$ \( 1 + 5 T^{2} + 14 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^2 \wr C_2$ \( 1 + 45 T^{2} + 834 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^2 \wr C_2$ \( 1 + 55 T^{2} + 1324 T^{4} + 55 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2 \wr C_2$ \( 1 + 18 T^{2} + 762 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 + 7 T + 60 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2^2 \wr C_2$ \( 1 + 68 T^{2} + 2422 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \)
37$D_{4}$ \( ( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^2 \wr C_2$ \( 1 + 108 T^{2} + 5622 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{4} \)
47$C_2^2 \wr C_2$ \( 1 + 7 T^{2} + 3928 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \)
53$D_{4}$ \( ( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^2 \wr C_2$ \( 1 + 2 T^{2} + 3642 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2 \wr C_2$ \( 1 + 138 T^{2} + 10194 T^{4} + 138 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{4} \)
71$D_{4}$ \( ( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2 \wr C_2$ \( 1 + 8 T^{2} + 10510 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2 \wr C_2$ \( 1 + 226 T^{2} + 24538 T^{4} + 226 p^{2} T^{6} + p^{4} T^{8} \)
89$C_2^2 \wr C_2$ \( 1 - 108 T^{2} + 16134 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2 \wr C_2$ \( 1 + 287 T^{2} + 38908 T^{4} + 287 p^{2} T^{6} + p^{4} T^{8} \)
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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

−5.56546580461882023139295778575, −5.11320351543683278825446588582, −5.03722029366731759008184705941, −4.95711658494598624678947483070, −4.76160256600655953992683224572, −4.27541290398350607731316516666, −4.17365423337434798287629933280, −4.02313333403403006430910951232, −3.95015247307215715082994888746, −3.78864310081288398013564822124, −3.39797257562260009540463345476, −3.37838304544697976849788569111, −3.37803160866828962358951579461, −2.68689120733514705272663148948, −2.65008767188382146097879888154, −2.54711345054479344072743172033, −2.50310813336746753780655070879, −1.86610295311175798520246165356, −1.85039770767340268923688672711, −1.83225397888700437354988471672, −1.32754865526699888660636141955, −0.963899144468201309344432732510, −0.70876267811370060081158803057, −0.59640830909633933612180879670, −0.40876501115478679853643355504, 0.40876501115478679853643355504, 0.59640830909633933612180879670, 0.70876267811370060081158803057, 0.963899144468201309344432732510, 1.32754865526699888660636141955, 1.83225397888700437354988471672, 1.85039770767340268923688672711, 1.86610295311175798520246165356, 2.50310813336746753780655070879, 2.54711345054479344072743172033, 2.65008767188382146097879888154, 2.68689120733514705272663148948, 3.37803160866828962358951579461, 3.37838304544697976849788569111, 3.39797257562260009540463345476, 3.78864310081288398013564822124, 3.95015247307215715082994888746, 4.02313333403403006430910951232, 4.17365423337434798287629933280, 4.27541290398350607731316516666, 4.76160256600655953992683224572, 4.95711658494598624678947483070, 5.03722029366731759008184705941, 5.11320351543683278825446588582, 5.56546580461882023139295778575

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