Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4-s + 5-s + 3·9-s + 4·11-s − 2·12-s − 2·15-s + 16-s + 20-s − 2·25-s − 4·27-s − 8·33-s + 3·36-s − 4·37-s + 4·44-s + 3·45-s + 16·47-s − 2·48-s + 2·49-s − 4·53-s + 4·55-s + 8·59-s − 2·60-s + 64-s − 8·67-s + 16·71-s + 4·75-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s + 0.447·5-s + 9-s + 1.20·11-s − 0.577·12-s − 0.516·15-s + 1/4·16-s + 0.223·20-s − 2/5·25-s − 0.769·27-s − 1.39·33-s + 1/2·36-s − 0.657·37-s + 0.603·44-s + 0.447·45-s + 2.33·47-s − 0.288·48-s + 2/7·49-s − 0.549·53-s + 0.539·55-s + 1.04·59-s − 0.258·60-s + 1/8·64-s − 0.977·67-s + 1.89·71-s + 0.461·75-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21780\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{21780} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 21780,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $1.049184077$
$L(\frac12)$  $\approx$  $1.049184077$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;5,\;11\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 2 T + p T^{2} ) \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$V_4$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$V_4$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$V_4$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$V_4$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$V_4$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−10.83812995438233265341033341820, −10.41137331151424430082490492183, −9.803064351421850226496301995439, −9.329172792456276416019531911476, −8.754931371176731833819107492896, −7.976194554773426566810206855329, −7.21201674050179372835015480714, −6.82378435606469854538163155443, −6.24843804240280296624777703161, −5.72945068915460235250490124046, −5.19439406411647697870324199977, −4.26893653779922028883425571731, −3.67813478878468384844838485841, −2.37891002330605190710787923901, −1.28513317176888455792170575842, 1.28513317176888455792170575842, 2.37891002330605190710787923901, 3.67813478878468384844838485841, 4.26893653779922028883425571731, 5.19439406411647697870324199977, 5.72945068915460235250490124046, 6.24843804240280296624777703161, 6.82378435606469854538163155443, 7.21201674050179372835015480714, 7.976194554773426566810206855329, 8.754931371176731833819107492896, 9.329172792456276416019531911476, 9.803064351421850226496301995439, 10.41137331151424430082490492183, 10.83812995438233265341033341820

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