Properties

Label 8-1950e4-1.1-c1e4-0-13
Degree $8$
Conductor $1.446\times 10^{13}$
Sign $1$
Analytic cond. $58782.3$
Root an. cond. $3.94598$
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 + 9-s + 10·11-s − 4·19-s + 10·29-s − 44·31-s + 36-s + 4·41-s + 10·44-s − 10·49-s − 30·59-s − 20·61-s − 64-s − 4·76-s + 44·79-s + 4·89-s + 10·99-s − 4·101-s + 8·109-s + 10·116-s + 47·121-s − 44·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s + 3.01·11-s − 0.917·19-s + 1.85·29-s − 7.90·31-s + 1/6·36-s + 0.624·41-s + 1.50·44-s − 1.42·49-s − 3.90·59-s − 2.56·61-s − 1/8·64-s − 0.458·76-s + 4.95·79-s + 0.423·89-s + 1.00·99-s − 0.398·101-s + 0.766·109-s + 0.928·116-s + 4.27·121-s − 3.95·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(58782.3\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.577369893\)
\(L(\frac12)\) \(\approx\) \(1.577369893\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - T^{2} + T^{4} \)
3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5 \( 1 \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
good7$C_2^3$ \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^2$$\times$$C_2^2$ \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \)
19$C_2^2$ \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2^2$ \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 11 T + p T^{2} )^{4} \)
37$C_2^3$ \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2^2$ \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2^3$ \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2^2$ \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2^2$ \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \)
67$C_2^2$$\times$$C_2^2$ \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \)
71$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^2$ \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^3$ \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \)
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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

−6.38983457656877956405881938589, −6.32800109421215922739135478849, −6.18384023069064505850465277195, −5.95518851020721700730465212473, −5.94449468473898000115201957124, −5.44100252246311032470146464377, −5.28457297368841068058511503464, −4.99424847125880081981777695669, −4.86609438457879286862394700778, −4.46293730273541636729741416599, −4.42898146892428771487688022695, −4.03454564218917444515837502544, −3.83444926623896607947559928947, −3.72492783203526042318530085444, −3.44365348809485501723750755906, −3.26240465982343213698943656388, −3.11107804089958557859208640472, −2.70326902081056409607779985780, −2.00283536386392535551068372740, −1.98340693993395811746211612441, −1.87958672546825060921466633712, −1.51260533180443386977805239735, −1.42570614934174808288392368187, −0.812420861447181400665992279812, −0.20962138138586612378228140035, 0.20962138138586612378228140035, 0.812420861447181400665992279812, 1.42570614934174808288392368187, 1.51260533180443386977805239735, 1.87958672546825060921466633712, 1.98340693993395811746211612441, 2.00283536386392535551068372740, 2.70326902081056409607779985780, 3.11107804089958557859208640472, 3.26240465982343213698943656388, 3.44365348809485501723750755906, 3.72492783203526042318530085444, 3.83444926623896607947559928947, 4.03454564218917444515837502544, 4.42898146892428771487688022695, 4.46293730273541636729741416599, 4.86609438457879286862394700778, 4.99424847125880081981777695669, 5.28457297368841068058511503464, 5.44100252246311032470146464377, 5.94449468473898000115201957124, 5.95518851020721700730465212473, 6.18384023069064505850465277195, 6.32800109421215922739135478849, 6.38983457656877956405881938589

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