Properties

Label 4-891e2-1.1-c1e2-0-18
Degree $4$
Conductor $793881$
Sign $1$
Analytic cond. $50.6185$
Root an. cond. $2.66733$
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  + 2-s + 2·4-s + 4·5-s + 2·7-s + 5·8-s + 4·10-s + 11-s + 2·13-s + 2·14-s + 5·16-s + 4·17-s − 12·19-s + 8·20-s + 22-s − 4·23-s + 5·25-s + 2·26-s + 4·28-s + 6·29-s − 4·31-s + 10·32-s + 4·34-s + 8·35-s − 12·37-s − 12·38-s + 20·40-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 4-s + 1.78·5-s + 0.755·7-s + 1.76·8-s + 1.26·10-s + 0.301·11-s + 0.554·13-s + 0.534·14-s + 5/4·16-s + 0.970·17-s − 2.75·19-s + 1.78·20-s + 0.213·22-s − 0.834·23-s + 25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 1.76·32-s + 0.685·34-s + 1.35·35-s − 1.97·37-s − 1.94·38-s + 3.16·40-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 793881 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(793881\)    =    \(3^{8} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(50.6185\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{891} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 793881,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.745359457\)
\(L(\frac12)\) \(\approx\) \(6.745359457\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44075119996207507133698932632, −10.09706287191021982127093132817, −9.734402582732179603870549351118, −8.968122324174631836059259933793, −8.594974867359990464099976452449, −8.371190256571500973490244011435, −7.51418003949278447526066592128, −7.47986182454641841972791260896, −6.64514109159385122075729483782, −6.34102406704256646259256177267, −6.08642031414999280199076545080, −5.50832957864501331384940877947, −5.19119003109069709016037664313, −4.50573274750858261865635783601, −4.14196833507790860893510165511, −3.64336894141285157711146419662, −2.63840021607510864531336217487, −2.14875707598864274370920681680, −1.80651653849649046040952359228, −1.28249196701563611811221350009, 1.28249196701563611811221350009, 1.80651653849649046040952359228, 2.14875707598864274370920681680, 2.63840021607510864531336217487, 3.64336894141285157711146419662, 4.14196833507790860893510165511, 4.50573274750858261865635783601, 5.19119003109069709016037664313, 5.50832957864501331384940877947, 6.08642031414999280199076545080, 6.34102406704256646259256177267, 6.64514109159385122075729483782, 7.47986182454641841972791260896, 7.51418003949278447526066592128, 8.371190256571500973490244011435, 8.594974867359990464099976452449, 8.968122324174631836059259933793, 9.734402582732179603870549351118, 10.09706287191021982127093132817, 10.44075119996207507133698932632

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