Properties

Label 4-888e2-1.1-c1e2-0-43
Degree $4$
Conductor $788544$
Sign $1$
Analytic cond. $50.2782$
Root an. cond. $2.66283$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 4·7-s + 3·9-s − 4·13-s − 8·15-s − 4·17-s − 8·21-s − 8·23-s + 4·25-s + 4·27-s − 12·29-s − 8·31-s + 16·35-s − 2·37-s − 8·39-s − 12·41-s − 12·45-s + 6·49-s − 8·51-s − 20·53-s + 4·61-s − 12·63-s + 16·65-s + 4·67-s − 16·69-s + 16·71-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 1.51·7-s + 9-s − 1.10·13-s − 2.06·15-s − 0.970·17-s − 1.74·21-s − 1.66·23-s + 4/5·25-s + 0.769·27-s − 2.22·29-s − 1.43·31-s + 2.70·35-s − 0.328·37-s − 1.28·39-s − 1.87·41-s − 1.78·45-s + 6/7·49-s − 1.12·51-s − 2.74·53-s + 0.512·61-s − 1.51·63-s + 1.98·65-s + 0.488·67-s − 1.92·69-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 788544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(788544\)    =    \(2^{6} \cdot 3^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(50.2782\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 788544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
37$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_m
7$C_4$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_k
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_w
17$D_{4}$ \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bk
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$D_{4}$ \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_ci
29$D_{4}$ \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_cy
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_cs
41$C_4$ \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.41.m_di
43$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \) 2.43.a_da
47$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.47.a_ck
53$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.53.u_hq
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_ac
67$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_fa
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.73.aq_ic
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.79.aq_hi
83$D_{4}$ \( 1 - 16 T + 158 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.83.aq_gc
89$D_{4}$ \( 1 + 12 T + 164 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gi
97$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_ac
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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

−9.605708302810561857543857926625, −9.579507885627775546323840448126, −9.195680100069341484459232340047, −8.631773565262735651496500130349, −7.962199603608401979576509164204, −7.958712784038577791958502206703, −7.52653331330372259265694680402, −7.06922861097897143725260519758, −6.49361818465513172144524720539, −6.39873199918971056593385782247, −5.31790321228225452677182245349, −5.04871420234573366796349749819, −4.11840636468448025348553360054, −3.93498839307269095925174637778, −3.38288832110874083387379337877, −3.33144520256563759123046508682, −2.13293635491393479106156780994, −2.04585992883107583908744567298, 0, 0, 2.04585992883107583908744567298, 2.13293635491393479106156780994, 3.33144520256563759123046508682, 3.38288832110874083387379337877, 3.93498839307269095925174637778, 4.11840636468448025348553360054, 5.04871420234573366796349749819, 5.31790321228225452677182245349, 6.39873199918971056593385782247, 6.49361818465513172144524720539, 7.06922861097897143725260519758, 7.52653331330372259265694680402, 7.958712784038577791958502206703, 7.962199603608401979576509164204, 8.631773565262735651496500130349, 9.195680100069341484459232340047, 9.579507885627775546323840448126, 9.605708302810561857543857926625

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