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Properties

Label 4-1224e2-1.1-c0e2-0-6
Degree $4$
Conductor $1498176$
Sign $1$
Analytic cond. $0.373144$
Root an. cond. $0.781572$
Motivic weight $0$
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  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 6·32-s − 4·34-s − 4·43-s + 7·64-s − 6·68-s − 8·86-s − 4·89-s + 2·121-s + 127-s + 8·128-s + 131-s − 8·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 12·172-s + 173-s + ⋯
L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 2·17-s + 6·32-s − 4·34-s − 4·43-s + 7·64-s − 6·68-s − 8·86-s − 4·89-s + 2·121-s + 127-s + 8·128-s + 131-s − 8·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s − 12·172-s + 173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1498176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1498176\)    =    \(2^{6} \cdot 3^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(0.373144\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1498176,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.805218009\)
\(L(\frac12)\) \(\approx\) \(3.805218009\)
\(L(1)\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
17$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + T^{4} \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2$ \( ( 1 + T^{2} )^{2} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$ \( ( 1 + T )^{4} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2^2$ \( 1 + T^{4} \)
73$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$ \( ( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−10.33179041141666961797126432776, −9.815145473967708308089653157924, −9.544650187060847345126315208888, −8.692049428683145076537394396596, −8.243467877985310963258160593262, −8.236147829360422414487098554936, −7.25659335317856591437310971845, −7.02130856197333576009564649286, −6.81684534549069567740192737690, −6.20225730906701865887343403843, −6.02025070473538959954755473884, −5.32645490573514714462000778054, −4.86477983149203238994211494033, −4.72989140648601119613823662494, −4.00707859043944259275973498021, −3.78640082738634303826746387055, −3.00000067620691898671501287645, −2.74675395656994056843698112633, −1.90424019366707809249124261889, −1.62287306692877604432792909434, 1.62287306692877604432792909434, 1.90424019366707809249124261889, 2.74675395656994056843698112633, 3.00000067620691898671501287645, 3.78640082738634303826746387055, 4.00707859043944259275973498021, 4.72989140648601119613823662494, 4.86477983149203238994211494033, 5.32645490573514714462000778054, 6.02025070473538959954755473884, 6.20225730906701865887343403843, 6.81684534549069567740192737690, 7.02130856197333576009564649286, 7.25659335317856591437310971845, 8.236147829360422414487098554936, 8.243467877985310963258160593262, 8.692049428683145076537394396596, 9.544650187060847345126315208888, 9.815145473967708308089653157924, 10.33179041141666961797126432776

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