Properties

Degree $4$
Conductor $1806336$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 5·7-s + 6·11-s + 6·13-s + 2·15-s − 4·17-s + 5·19-s − 5·21-s − 4·23-s + 5·25-s − 27-s + 8·29-s + 7·31-s + 6·33-s − 10·35-s − 9·37-s + 6·39-s − 4·41-s − 2·43-s + 2·47-s + 18·49-s − 4·51-s + 8·53-s + 12·55-s + 5·57-s + 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 1.88·7-s + 1.80·11-s + 1.66·13-s + 0.516·15-s − 0.970·17-s + 1.14·19-s − 1.09·21-s − 0.834·23-s + 25-s − 0.192·27-s + 1.48·29-s + 1.25·31-s + 1.04·33-s − 1.69·35-s − 1.47·37-s + 0.960·39-s − 0.624·41-s − 0.304·43-s + 0.291·47-s + 18/7·49-s − 0.560·51-s + 1.09·53-s + 1.61·55-s + 0.662·57-s + 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1806336\)    =    \(2^{12} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1344} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1806336,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.527317695\)
\(L(\frac12)\) \(\approx\) \(3.527317695\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 9 T + 44 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 8 T + 11 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 15 T + 158 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−9.661828778204744979263752974985, −9.454157858589829136293294036860, −9.061147635327130878898546492572, −8.618819852881984684475716843559, −8.476886523323609540548834333354, −7.996967516213156089272758234472, −7.05527745628719582604363649878, −6.69569905015701800945830599205, −6.60166636060455075415403691746, −6.35577448308589442350525109577, −5.78321394563122959727903046208, −5.33631258717304183008560037926, −4.71496711532365541581977928842, −3.87557989379165919252853137159, −3.69622942818843229777735936602, −3.43769372299176148214603165299, −2.62488581165896639895207039758, −2.29084217163218853439282551178, −1.31926011431250485134687295627, −0.846566050412001695512001384867, 0.846566050412001695512001384867, 1.31926011431250485134687295627, 2.29084217163218853439282551178, 2.62488581165896639895207039758, 3.43769372299176148214603165299, 3.69622942818843229777735936602, 3.87557989379165919252853137159, 4.71496711532365541581977928842, 5.33631258717304183008560037926, 5.78321394563122959727903046208, 6.35577448308589442350525109577, 6.60166636060455075415403691746, 6.69569905015701800945830599205, 7.05527745628719582604363649878, 7.996967516213156089272758234472, 8.476886523323609540548834333354, 8.618819852881984684475716843559, 9.061147635327130878898546492572, 9.454157858589829136293294036860, 9.661828778204744979263752974985

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