Properties

Label 4-21780-1.1-c1e2-0-0
Degree $4$
Conductor $21780$
Sign $1$
Analytic cond. $1.38871$
Root an. cond. $1.08555$
Motivic weight $1$
Arithmetic yes
Rational yes
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

Degree: \(4\)
Conductor: \(21780\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1.38871\)
Root analytic conductor: \(1.08555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21780,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.049184077\)
\(L(\frac12)\) \(\approx\) \(1.049184077\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
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$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
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$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−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