Properties

Label 4-864e2-1.1-c1e2-0-16
Degree $4$
Conductor $746496$
Sign $1$
Analytic cond. $47.5972$
Root an. cond. $2.62660$
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  − 2·11-s + 8·13-s + 12·23-s + 3·25-s + 20·37-s + 20·47-s − 49-s + 8·59-s − 16·71-s − 6·73-s − 18·83-s + 14·97-s − 34·107-s + 4·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s − 16·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯
L(s)  = 1  − 0.603·11-s + 2.21·13-s + 2.50·23-s + 3/5·25-s + 3.28·37-s + 2.91·47-s − 1/7·49-s + 1.04·59-s − 1.89·71-s − 0.702·73-s − 1.97·83-s + 1.42·97-s − 3.28·107-s + 0.383·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.33·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(746496\)    =    \(2^{10} \cdot 3^{6}\)
Sign: $1$
Analytic conductor: \(47.5972\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{864} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 746496,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.621732570\)
\(L(\frac12)\) \(\approx\) \(2.621732570\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 49 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 93 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−10.61329130962769716821788550023, −9.964840343938564478644416184463, −9.452113455488338039701609058853, −8.928035848704536448691635188715, −8.830254827021835108430879729823, −8.393739727683504466634359674695, −7.78611273786181910275629725402, −7.44872784047950347991868238866, −6.95068919942131121681486334724, −6.49335744037121874476973366301, −5.89942799257367761273399751420, −5.72670324801817356906859602297, −5.12993293925237691320274525243, −4.44082148729733690222942083924, −4.13218273163520837512685375835, −3.49977165664856287199112412591, −2.68604938373979088836071886122, −2.67852308830331863665695825763, −1.23059609338931193515469368251, −0.989698206489348970590235414714, 0.989698206489348970590235414714, 1.23059609338931193515469368251, 2.67852308830331863665695825763, 2.68604938373979088836071886122, 3.49977165664856287199112412591, 4.13218273163520837512685375835, 4.44082148729733690222942083924, 5.12993293925237691320274525243, 5.72670324801817356906859602297, 5.89942799257367761273399751420, 6.49335744037121874476973366301, 6.95068919942131121681486334724, 7.44872784047950347991868238866, 7.78611273786181910275629725402, 8.393739727683504466634359674695, 8.830254827021835108430879729823, 8.928035848704536448691635188715, 9.452113455488338039701609058853, 9.964840343938564478644416184463, 10.61329130962769716821788550023

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