Properties

Degree $4$
Conductor $1072476$
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  + 4-s − 3·9-s + 16-s − 12·17-s + 8·19-s + 16·23-s − 6·25-s + 4·29-s − 31-s − 3·36-s − 14·49-s − 12·53-s + 64-s − 24·67-s − 12·68-s + 8·76-s + 9·81-s + 16·83-s − 12·89-s + 16·92-s + 4·97-s − 6·100-s + 16·103-s − 4·109-s + 4·116-s − 22·121-s − 124-s + ⋯
L(s)  = 1  + 1/2·4-s − 9-s + 1/4·16-s − 2.91·17-s + 1.83·19-s + 3.33·23-s − 6/5·25-s + 0.742·29-s − 0.179·31-s − 1/2·36-s − 2·49-s − 1.64·53-s + 1/8·64-s − 2.93·67-s − 1.45·68-s + 0.917·76-s + 81-s + 1.75·83-s − 1.27·89-s + 1.66·92-s + 0.406·97-s − 3/5·100-s + 1.57·103-s − 0.383·109-s + 0.371·116-s − 2·121-s − 0.0898·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1072476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1072476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1072476\)    =    \(2^{2} \cdot 3^{2} \cdot 31^{3}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{1072476} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1072476,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + p T^{2} \)
31$C_1$ \( 1 + T \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.76857292337562945056886541715, −7.43556895569719724485579804346, −6.93807340940953532824726950400, −6.48842833802306821880895380523, −6.32858579420662158347624019611, −5.61110203252813853267502993088, −5.11987644824288757943667009350, −4.74203912193695697715499891246, −4.37664321587824825312436918660, −3.29788473612982991726310940500, −3.16188554299090604324575557253, −2.63802468934373486533184492979, −1.92931953630680217049542026766, −1.15997339964107602320819529739, 0, 1.15997339964107602320819529739, 1.92931953630680217049542026766, 2.63802468934373486533184492979, 3.16188554299090604324575557253, 3.29788473612982991726310940500, 4.37664321587824825312436918660, 4.74203912193695697715499891246, 5.11987644824288757943667009350, 5.61110203252813853267502993088, 6.32858579420662158347624019611, 6.48842833802306821880895380523, 6.93807340940953532824726950400, 7.43556895569719724485579804346, 7.76857292337562945056886541715

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