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Properties

Label 4-777e2-1.1-c1e2-0-33
Degree $4$
Conductor $603729$
Sign $1$
Analytic cond. $38.4942$
Root an. cond. $2.49085$
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  + 2·3-s + 4-s − 2·7-s + 3·9-s + 2·12-s − 3·16-s − 4·21-s + 10·25-s + 4·27-s − 2·28-s + 3·36-s − 10·37-s + 12·41-s − 6·48-s + 3·49-s + 12·53-s − 6·63-s − 7·64-s + 8·67-s + 24·71-s − 28·73-s + 20·75-s + 5·81-s − 24·83-s − 4·84-s + 10·100-s + 36·101-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 0.755·7-s + 9-s + 0.577·12-s − 3/4·16-s − 0.872·21-s + 2·25-s + 0.769·27-s − 0.377·28-s + 1/2·36-s − 1.64·37-s + 1.87·41-s − 0.866·48-s + 3/7·49-s + 1.64·53-s − 0.755·63-s − 7/8·64-s + 0.977·67-s + 2.84·71-s − 3.27·73-s + 2.30·75-s + 5/9·81-s − 2.63·83-s − 0.436·84-s + 100-s + 3.58·101-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(603729\)    =    \(3^{2} \cdot 7^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(38.4942\)
Root analytic conductor: \(2.49085\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 603729,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.168865726\)
\(L(\frac12)\) \(\approx\) \(3.168865726\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.31.a_aby
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.59.a_aec
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.73.bc_ne
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.89.a_afa
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \) 2.97.a_afq
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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

−10.49699898659939797420674277281, −10.11665418914636478050690839411, −9.544943459220891083746956718582, −9.206614446968691492852702912865, −8.713139450412145341897346520697, −8.634276120067444558573660470230, −8.020519411130146054614365240305, −7.24161506675600663556993478998, −7.22695380700277982031539082137, −6.76666887233713438087991864066, −6.31228952920442840308811741312, −5.66441572403190959902235634610, −5.16430464129735883637214474798, −4.41446742503436189509795148223, −4.13342641843724607408356055279, −3.31176843234929359443587915644, −3.01064210701561253546048296174, −2.44119742557556288003545275496, −1.88299682718152455224553913205, −0.851756530858945826244410574704, 0.851756530858945826244410574704, 1.88299682718152455224553913205, 2.44119742557556288003545275496, 3.01064210701561253546048296174, 3.31176843234929359443587915644, 4.13342641843724607408356055279, 4.41446742503436189509795148223, 5.16430464129735883637214474798, 5.66441572403190959902235634610, 6.31228952920442840308811741312, 6.76666887233713438087991864066, 7.22695380700277982031539082137, 7.24161506675600663556993478998, 8.020519411130146054614365240305, 8.634276120067444558573660470230, 8.713139450412145341897346520697, 9.206614446968691492852702912865, 9.544943459220891083746956718582, 10.11665418914636478050690839411, 10.49699898659939797420674277281

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