Properties

Label 4-42e2-1.1-c1e2-0-2
Degree $4$
Conductor $1764$
Sign $1$
Analytic cond. $0.112474$
Root an. cond. $0.579112$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 3·5-s − 6-s + 5·7-s − 8-s − 3·10-s − 3·11-s − 8·13-s + 5·14-s + 3·15-s − 16-s + 4·19-s − 5·21-s − 3·22-s + 24-s + 5·25-s − 8·26-s + 27-s + 18·29-s + 3·30-s + 31-s + 3·33-s − 15·35-s − 8·37-s + 4·38-s + 8·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1.34·5-s − 0.408·6-s + 1.88·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s − 2.21·13-s + 1.33·14-s + 0.774·15-s − 1/4·16-s + 0.917·19-s − 1.09·21-s − 0.639·22-s + 0.204·24-s + 25-s − 1.56·26-s + 0.192·27-s + 3.34·29-s + 0.547·30-s + 0.179·31-s + 0.522·33-s − 2.53·35-s − 1.31·37-s + 0.648·38-s + 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.112474\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{42} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1764,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6404112297\)
\(L(\frac12)\) \(\approx\) \(0.6404112297\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
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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

−16.17973659941015087962770093684, −15.60313265224890188290418727783, −15.21881119297981133878847469119, −14.61225741559721017060439804948, −14.16408823459251744051413936216, −13.65356895603610477074427325375, −12.56353919645869054505588797252, −12.04589492707314458522553361968, −11.81984739532897100902202830412, −11.38368210113026558031237446692, −10.34551278070132658000810499190, −10.08359043663864915519021896089, −8.508374544210305527887034814605, −8.258208070285795497623176922352, −7.43633156487582478996632849091, −6.89022930108548085633542082608, −5.17981846056775804588147047858, −5.05183042568136278786904085559, −4.39806420206531623293129256174, −2.83236694053324174912444359948, 2.83236694053324174912444359948, 4.39806420206531623293129256174, 5.05183042568136278786904085559, 5.17981846056775804588147047858, 6.89022930108548085633542082608, 7.43633156487582478996632849091, 8.258208070285795497623176922352, 8.508374544210305527887034814605, 10.08359043663864915519021896089, 10.34551278070132658000810499190, 11.38368210113026558031237446692, 11.81984739532897100902202830412, 12.04589492707314458522553361968, 12.56353919645869054505588797252, 13.65356895603610477074427325375, 14.16408823459251744051413936216, 14.61225741559721017060439804948, 15.21881119297981133878847469119, 15.60313265224890188290418727783, 16.17973659941015087962770093684

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