Properties

Degree $4$
Conductor $446224$
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 − 6·5-s + 3·7-s − 2·9-s + 9·13-s − 6·15-s + 17-s + 4·19-s + 3·21-s − 23-s + 17·25-s − 2·27-s + 8·29-s + 6·31-s − 18·35-s + 8·37-s + 9·39-s + 2·41-s + 12·43-s + 12·45-s − 4·47-s − 4·49-s + 51-s − 2·53-s + 4·57-s − 6·59-s − 8·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 2.68·5-s + 1.13·7-s − 2/3·9-s + 2.49·13-s − 1.54·15-s + 0.242·17-s + 0.917·19-s + 0.654·21-s − 0.208·23-s + 17/5·25-s − 0.384·27-s + 1.48·29-s + 1.07·31-s − 3.04·35-s + 1.31·37-s + 1.44·39-s + 0.312·41-s + 1.82·43-s + 1.78·45-s − 0.583·47-s − 4/7·49-s + 0.140·51-s − 0.274·53-s + 0.529·57-s − 0.781·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 446224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(446224\)    =    \(2^{4} \cdot 167^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{668} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 446224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72944\)
\(L(\frac12)\) \(\approx\) \(1.72944\)
\(L(\frac{3}{2})\) 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 \( 1 \)
167$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
7$D_{4}$ \( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 9 T + 43 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 31 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 77 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 2 T + 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 109 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 85 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 94 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 21 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 115 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 20 T + 245 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 157 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 70 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 133 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−10.78684176606250352780348090690, −10.73181878433097000391309121198, −9.808267262921383522661575283509, −9.220792238273381390054638834383, −8.671315156529340627781238131251, −8.476280579730050832206196218228, −8.054853667853943176173593229555, −7.82032309207390708230357363588, −7.65098058816185145766008554081, −6.90567599531242657940243573298, −6.10159033130350986165324583392, −6.05745921150348059150766290230, −5.02128472225635093282454747163, −4.59745577066120010943531232006, −3.99613816093496145148275393777, −3.87332638670442420979740841884, −2.98394981789951374605396405234, −2.94107401849814219392407337671, −1.41901172003878071098671570951, −0.78301635456452129501850509076, 0.78301635456452129501850509076, 1.41901172003878071098671570951, 2.94107401849814219392407337671, 2.98394981789951374605396405234, 3.87332638670442420979740841884, 3.99613816093496145148275393777, 4.59745577066120010943531232006, 5.02128472225635093282454747163, 6.05745921150348059150766290230, 6.10159033130350986165324583392, 6.90567599531242657940243573298, 7.65098058816185145766008554081, 7.82032309207390708230357363588, 8.054853667853943176173593229555, 8.476280579730050832206196218228, 8.671315156529340627781238131251, 9.220792238273381390054638834383, 9.808267262921383522661575283509, 10.73181878433097000391309121198, 10.78684176606250352780348090690

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