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Properties

Label 4-888e2-1.1-c1e2-0-33
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  − 2·2-s + 2·4-s − 4·5-s − 9-s + 8·10-s − 2·13-s − 4·16-s − 4·17-s + 2·18-s + 2·19-s − 8·20-s − 4·23-s + 8·25-s + 4·26-s − 4·29-s − 14·31-s + 8·32-s + 8·34-s − 2·36-s − 2·37-s − 4·38-s + 6·43-s + 4·45-s + 8·46-s + 10·49-s − 16·50-s − 4·52-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 1.78·5-s − 1/3·9-s + 2.52·10-s − 0.554·13-s − 16-s − 0.970·17-s + 0.471·18-s + 0.458·19-s − 1.78·20-s − 0.834·23-s + 8/5·25-s + 0.784·26-s − 0.742·29-s − 2.51·31-s + 1.41·32-s + 1.37·34-s − 1/3·36-s − 0.328·37-s − 0.648·38-s + 0.914·43-s + 0.596·45-s + 1.17·46-s + 10/7·49-s − 2.26·50-s − 0.554·52-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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + T^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.5.e_i
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.c_c
17$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_i
19$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_c
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_i
29$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_i
31$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.31.o_du
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.41.a_ada
43$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_s
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.59.q_ey
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_s
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + p^{2} T^{4} \) 2.89.a_a
97$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_s
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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

−9.738633219224020670869408153345, −9.321924135994664025758133813425, −9.108133118196141956323796647472, −8.720641910488239256617846959335, −8.099025046418269072891519114432, −7.981126261820416233182584945004, −7.34847979860524319173576549594, −7.27314193888750601323625848807, −6.91534628999212558492856793582, −6.06767403421371306206175065841, −5.66648279770162576168985679613, −4.91607225912987517143439621329, −4.50867558539335250968776920496, −3.78090169944807331430310030282, −3.73035769981333388573314648440, −2.72766549364153072777069575179, −2.15333976409699628574921211347, −1.33002203539019968278374339651, 0, 0, 1.33002203539019968278374339651, 2.15333976409699628574921211347, 2.72766549364153072777069575179, 3.73035769981333388573314648440, 3.78090169944807331430310030282, 4.50867558539335250968776920496, 4.91607225912987517143439621329, 5.66648279770162576168985679613, 6.06767403421371306206175065841, 6.91534628999212558492856793582, 7.27314193888750601323625848807, 7.34847979860524319173576549594, 7.981126261820416233182584945004, 8.099025046418269072891519114432, 8.720641910488239256617846959335, 9.108133118196141956323796647472, 9.321924135994664025758133813425, 9.738633219224020670869408153345

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