Properties

Label 4-435e2-1.1-c1e2-0-2
Degree $4$
Conductor $189225$
Sign $1$
Analytic cond. $12.0651$
Root an. cond. $1.86373$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 2·3-s + 8·4-s + 4·5-s − 8·6-s − 8·8-s + 3·9-s − 16·10-s + 16·12-s + 8·15-s − 4·16-s − 12·17-s − 12·18-s + 32·20-s − 16·24-s + 11·25-s + 4·27-s − 4·29-s − 32·30-s + 32·32-s + 48·34-s + 24·36-s − 2·37-s − 32·40-s − 2·43-s + 12·45-s + 16·47-s + ⋯
L(s)  = 1  − 2.82·2-s + 1.15·3-s + 4·4-s + 1.78·5-s − 3.26·6-s − 2.82·8-s + 9-s − 5.05·10-s + 4.61·12-s + 2.06·15-s − 16-s − 2.91·17-s − 2.82·18-s + 7.15·20-s − 3.26·24-s + 11/5·25-s + 0.769·27-s − 0.742·29-s − 5.84·30-s + 5.65·32-s + 8.23·34-s + 4·36-s − 0.328·37-s − 5.05·40-s − 0.304·43-s + 1.78·45-s + 2.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(189225\)    =    \(3^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(12.0651\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 189225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7695478471\)
\(L(\frac12)\) \(\approx\) \(0.7695478471\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
29$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + p T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 174 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
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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

−10.97354914286742907840547048921, −10.64376853280538732873835196580, −10.16611440680651221428335497561, −9.825559453686683173643219770104, −9.307911693400769796865292465243, −9.158389456161346471318345142439, −8.875556592568695344666217360988, −8.435083951764939988995447218328, −8.095690002938542576854341536020, −7.38245951446731992835070226902, −6.82866517287244109916693696737, −6.79568999643739799447282039592, −6.09095042061882719816334988417, −5.15598333046146503472668996480, −4.57114576488874229321384665906, −3.85144277930253994974000303287, −2.52509541441722889404633038325, −2.11783780223542915033328418052, −1.96327910702602339931425882472, −0.834978617783562150213868522314, 0.834978617783562150213868522314, 1.96327910702602339931425882472, 2.11783780223542915033328418052, 2.52509541441722889404633038325, 3.85144277930253994974000303287, 4.57114576488874229321384665906, 5.15598333046146503472668996480, 6.09095042061882719816334988417, 6.79568999643739799447282039592, 6.82866517287244109916693696737, 7.38245951446731992835070226902, 8.095690002938542576854341536020, 8.435083951764939988995447218328, 8.875556592568695344666217360988, 9.158389456161346471318345142439, 9.307911693400769796865292465243, 9.825559453686683173643219770104, 10.16611440680651221428335497561, 10.64376853280538732873835196580, 10.97354914286742907840547048921

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