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Properties

Label 4-777e2-1.1-c0e2-0-0
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $0.150368$
Root an. cond. $0.622714$
Motivic weight $0$
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  − 2·5-s + 2·7-s − 9-s − 16-s + 2·17-s + 2·25-s − 4·35-s − 2·43-s + 2·45-s + 3·49-s + 2·59-s − 2·63-s + 2·79-s + 2·80-s + 81-s + 4·83-s − 4·85-s − 2·89-s − 2·109-s − 2·112-s + 4·119-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 2·5-s + 2·7-s − 9-s − 16-s + 2·17-s + 2·25-s − 4·35-s − 2·43-s + 2·45-s + 3·49-s + 2·59-s − 2·63-s + 2·79-s + 2·80-s + 81-s + 4·83-s − 4·85-s − 2·89-s − 2·109-s − 2·112-s + 4·119-s − 2·121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(603729\)    =    \(3^{2} \cdot 7^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.150368\)
Root analytic conductor: \(0.622714\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7153961835\)
\(L(\frac12)\) \(\approx\) \(0.7153961835\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
37$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
5$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2$ \( ( 1 + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_2^2$ \( 1 + T^{4} \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
83$C_1$ \( ( 1 - T )^{4} \)
89$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−11.05642525438482019890253504223, −10.36083690280556504892501268837, −10.09045604760130454787215968241, −9.098721141271697609114017869113, −9.049972179239141072564591541794, −8.341837928933393863668457480641, −8.039470672228302394721514508922, −7.77701095497476555853585799928, −7.74009376506585359357403103394, −6.75746028764500147447942077355, −6.68779108672136037735494074000, −5.59584092659544347351645893187, −5.21715438124194475900243651267, −5.01584006950340574836907555228, −4.32853803433833152787050427985, −3.82398391179252951417187775505, −3.47225769203919290802917903174, −2.71629453829185550282097748806, −1.95198756785957959256130349811, −0.965207469623200946542639481084, 0.965207469623200946542639481084, 1.95198756785957959256130349811, 2.71629453829185550282097748806, 3.47225769203919290802917903174, 3.82398391179252951417187775505, 4.32853803433833152787050427985, 5.01584006950340574836907555228, 5.21715438124194475900243651267, 5.59584092659544347351645893187, 6.68779108672136037735494074000, 6.75746028764500147447942077355, 7.74009376506585359357403103394, 7.77701095497476555853585799928, 8.039470672228302394721514508922, 8.341837928933393863668457480641, 9.049972179239141072564591541794, 9.098721141271697609114017869113, 10.09045604760130454787215968241, 10.36083690280556504892501268837, 11.05642525438482019890253504223

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