Properties

Degree $4$
Conductor $609408$
Sign $-1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 9-s − 8·11-s − 4·13-s + 8·19-s + 8·23-s − 6·25-s + 12·29-s − 12·41-s − 8·43-s − 14·49-s + 8·67-s + 20·73-s + 16·79-s + 81-s + 8·83-s − 8·99-s − 36·101-s − 32·103-s + 24·107-s − 4·117-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 32·143-s + 149-s + ⋯
L(s)  = 1  + 1/3·9-s − 2.41·11-s − 1.10·13-s + 1.83·19-s + 1.66·23-s − 6/5·25-s + 2.22·29-s − 1.87·41-s − 1.21·43-s − 2·49-s + 0.977·67-s + 2.34·73-s + 1.80·79-s + 1/9·81-s + 0.878·83-s − 0.804·99-s − 3.58·101-s − 3.15·103-s + 2.32·107-s − 0.369·117-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.67·143-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(609408\)    =    \(2^{7} \cdot 3^{2} \cdot 23^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{609408} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 609408,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
23$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.173282349834764667604075089133, −7.932741438944089455498075047781, −7.26958585488936859387169650645, −6.81091891714363813978370387900, −6.61894821311647461036273237236, −5.68693030173540232457460232898, −5.08637437240091893034392278396, −5.01357758335939619070346679818, −4.82931243251591807244664468719, −3.66511529305920391284937215509, −3.14826068230388029805668446658, −2.74128373145577575704343979657, −2.15578307237422591548694616714, −1.13850789947910914060996228408, 0, 1.13850789947910914060996228408, 2.15578307237422591548694616714, 2.74128373145577575704343979657, 3.14826068230388029805668446658, 3.66511529305920391284937215509, 4.82931243251591807244664468719, 5.01357758335939619070346679818, 5.08637437240091893034392278396, 5.68693030173540232457460232898, 6.61894821311647461036273237236, 6.81091891714363813978370387900, 7.26958585488936859387169650645, 7.932741438944089455498075047781, 8.173282349834764667604075089133

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