Properties

Degree $4$
Conductor $1115136$
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  + 2·3-s + 3·9-s − 4·11-s − 4·13-s − 6·25-s + 4·27-s + 12·29-s − 8·33-s − 8·39-s − 14·49-s − 8·59-s − 4·61-s + 8·67-s − 12·75-s + 16·79-s + 5·81-s + 24·87-s − 12·89-s + 4·97-s − 12·99-s − 36·101-s − 4·109-s + 36·113-s − 12·117-s + 5·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 1.15·3-s + 9-s − 1.20·11-s − 1.10·13-s − 6/5·25-s + 0.769·27-s + 2.22·29-s − 1.39·33-s − 1.28·39-s − 2·49-s − 1.04·59-s − 0.512·61-s + 0.977·67-s − 1.38·75-s + 1.80·79-s + 5/9·81-s + 2.57·87-s − 1.27·89-s + 0.406·97-s − 1.20·99-s − 3.58·101-s − 0.383·109-s + 3.38·113-s − 1.10·117-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1115136\)    =    \(2^{10} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1115136} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1115136,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−8.110766320247241616629975072243, −7.36161606524251319117169763439, −7.26958585488936859387169650645, −6.55322332795183478795265868690, −6.22453934394527685809407419326, −5.53159840155631813382842710781, −5.01357758335939619070346679818, −4.62970445405048247550620463753, −4.26505152086983267813728153825, −3.36661049671147278939220747707, −3.14826068230388029805668446658, −2.42198212265018139467508199570, −2.20178933375822209510979578209, −1.26338131443797891405506654201, 0, 1.26338131443797891405506654201, 2.20178933375822209510979578209, 2.42198212265018139467508199570, 3.14826068230388029805668446658, 3.36661049671147278939220747707, 4.26505152086983267813728153825, 4.62970445405048247550620463753, 5.01357758335939619070346679818, 5.53159840155631813382842710781, 6.22453934394527685809407419326, 6.55322332795183478795265868690, 7.26958585488936859387169650645, 7.36161606524251319117169763439, 8.110766320247241616629975072243

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