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Properties

Label 4-888e2-1.1-c1e2-0-36
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 6·7-s − 2·9-s − 4·19-s + 6·21-s − 10·25-s + 5·27-s − 8·31-s + 2·37-s + 8·43-s + 13·49-s + 4·57-s − 8·61-s + 12·63-s + 24·67-s − 26·73-s + 10·75-s − 20·79-s + 81-s + 8·93-s − 24·97-s − 16·103-s − 28·109-s − 2·111-s − 13·121-s + 127-s − 8·129-s + ⋯
L(s)  = 1  − 0.577·3-s − 2.26·7-s − 2/3·9-s − 0.917·19-s + 1.30·21-s − 2·25-s + 0.962·27-s − 1.43·31-s + 0.328·37-s + 1.21·43-s + 13/7·49-s + 0.529·57-s − 1.02·61-s + 1.51·63-s + 2.93·67-s − 3.04·73-s + 1.15·75-s − 2.25·79-s + 1/9·81-s + 0.829·93-s − 2.43·97-s − 1.57·103-s − 2.68·109-s − 0.189·111-s − 1.18·121-s + 0.0887·127-s − 0.704·129-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$C_2$ \( 1 + T + p T^{2} \)
37$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
41$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.41.a_bh
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.47.a_dp
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.61.i_fi
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.71.a_en
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.73.ba_md
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.83.a_gj
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.97.y_na
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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

−7.65286120224110903388505491453, −7.34822351800845502523348597298, −6.72433556984559570880095854341, −6.40256795051194833967277103826, −6.08449220356234607481996960226, −5.54270171583374531541087657986, −5.43110991062316119422108197791, −4.38932651849542855280266113316, −3.93279504709150990560375020684, −3.59781441472728197557363900163, −2.76426174907496236443941498069, −2.58862548114280755949345221322, −1.50974967736814861270341338685, 0, 0, 1.50974967736814861270341338685, 2.58862548114280755949345221322, 2.76426174907496236443941498069, 3.59781441472728197557363900163, 3.93279504709150990560375020684, 4.38932651849542855280266113316, 5.43110991062316119422108197791, 5.54270171583374531541087657986, 6.08449220356234607481996960226, 6.40256795051194833967277103826, 6.72433556984559570880095854341, 7.34822351800845502523348597298, 7.65286120224110903388505491453

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