Properties

Label 8-70e8-1.1-c1e4-0-5
Degree $8$
Conductor $5.765\times 10^{14}$
Sign $1$
Analytic cond. $2.34364\times 10^{6}$
Root an. cond. $6.25513$
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  + 9-s + 6·11-s + 18·29-s − 48·71-s + 2·79-s + 9·81-s + 6·99-s + 22·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 19·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1/3·9-s + 1.80·11-s + 3.34·29-s − 5.69·71-s + 0.225·79-s + 81-s + 0.603·99-s + 2.10·109-s + 2.81·121-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 − 1.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.34364\times 10^{6}\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.557418768\)
\(L(\frac12)\) \(\approx\) \(8.557418768\)
\(L(\frac{3}{2})\) 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$C_2^3$ \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \)
13$C_2^3$ \( 1 + 19 T^{2} + 192 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2^3$ \( 1 - 29 T^{2} + 552 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2$ \( ( 1 + p T^{2} )^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{4} \)
29$C_2^2$ \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{4} \)
37$C_2$ \( ( 1 + p T^{2} )^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{4} \)
47$C_2^3$ \( 1 + 31 T^{2} - 1248 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2$ \( ( 1 + p T^{2} )^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{4} \)
73$C_2^2$ \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{4} \)
97$C_2^3$ \( 1 - 149 T^{2} + 12792 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.03664732574550034845785031768, −5.52459314424103765757153694479, −5.41442469145351755585461746783, −5.34648412163491855297776520707, −5.08844801340037788588471845931, −4.64881673006805474163027534472, −4.53490359577722223184028317113, −4.42866049153897732794754696209, −4.31281518764731600420055245092, −4.22890320034163812718308823143, −3.98856851767641320321434589048, −3.46284291671323414524726911125, −3.42053151735048252977608392448, −3.25806179496169161883615951599, −2.91686934021750974749645393946, −2.79021922724531219355238059065, −2.76352904329748997353639680725, −2.19893308165409539608203511014, −1.90613398862850428366013765502, −1.75447112156164549141202643939, −1.60186170130495128133513635298, −1.11891767210185429297561890902, −1.05584080599531582497237232883, −0.57441146660435581255081164576, −0.46638524026184820567513377285, 0.46638524026184820567513377285, 0.57441146660435581255081164576, 1.05584080599531582497237232883, 1.11891767210185429297561890902, 1.60186170130495128133513635298, 1.75447112156164549141202643939, 1.90613398862850428366013765502, 2.19893308165409539608203511014, 2.76352904329748997353639680725, 2.79021922724531219355238059065, 2.91686934021750974749645393946, 3.25806179496169161883615951599, 3.42053151735048252977608392448, 3.46284291671323414524726911125, 3.98856851767641320321434589048, 4.22890320034163812718308823143, 4.31281518764731600420055245092, 4.42866049153897732794754696209, 4.53490359577722223184028317113, 4.64881673006805474163027534472, 5.08844801340037788588471845931, 5.34648412163491855297776520707, 5.41442469145351755585461746783, 5.52459314424103765757153694479, 6.03664732574550034845785031768

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