Properties

Degree 4
Conductor $ 2^{2} \cdot 11^{2} \cdot 19^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·5-s − 3·9-s + 2·11-s + 16-s − 4·20-s + 10·23-s + 2·25-s − 3·36-s + 2·44-s + 12·45-s − 16·47-s − 5·49-s − 8·55-s + 64-s − 4·80-s + 10·92-s − 6·99-s + 2·100-s − 40·115-s − 7·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.78·5-s − 9-s + 0.603·11-s + 1/4·16-s − 0.894·20-s + 2.08·23-s + 2/5·25-s − 1/2·36-s + 0.301·44-s + 1.78·45-s − 2.33·47-s − 5/7·49-s − 1.07·55-s + 1/8·64-s − 0.447·80-s + 1.04·92-s − 0.603·99-s + 1/5·100-s − 3.73·115-s − 0.636·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/4·144-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(174724\)    =    \(2^{2} \cdot 11^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{174724} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 174724,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
19$C_2$ \( 1 + p T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.790588601535390277292601816676, −8.428795589558927348932545064837, −7.932250930921357396197811411568, −7.59637740604431091607700513464, −7.04729226132818219446518683825, −6.59351405053505432939215471518, −6.11693789079955114728164002848, −5.35115169010777942506357811198, −4.85738801044108780443732649673, −4.25517082503526883883452937780, −3.45956340591956291735517152818, −3.32009790808247759240534168188, −2.49392875440254451170783957070, −1.31654176033133009955997278519, 0, 1.31654176033133009955997278519, 2.49392875440254451170783957070, 3.32009790808247759240534168188, 3.45956340591956291735517152818, 4.25517082503526883883452937780, 4.85738801044108780443732649673, 5.35115169010777942506357811198, 6.11693789079955114728164002848, 6.59351405053505432939215471518, 7.04729226132818219446518683825, 7.59637740604431091607700513464, 7.932250930921357396197811411568, 8.428795589558927348932545064837, 8.790588601535390277292601816676

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