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Properties

Label 4-5440e2-1.1-c1e2-0-5
Degree $4$
Conductor $29593600$
Sign $1$
Analytic cond. $1886.91$
Root an. cond. $6.59079$
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 − 2·5-s − 2·7-s − 9-s − 8·11-s − 5·13-s + 2·15-s + 2·17-s − 19-s + 2·21-s + 2·23-s + 3·25-s − 29-s + 9·31-s + 8·33-s + 4·35-s + 6·37-s + 5·39-s − 8·41-s + 6·43-s + 2·45-s − 9·47-s + 6·49-s − 2·51-s − 3·53-s + 16·55-s + 57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s − 1/3·9-s − 2.41·11-s − 1.38·13-s + 0.516·15-s + 0.485·17-s − 0.229·19-s + 0.436·21-s + 0.417·23-s + 3/5·25-s − 0.185·29-s + 1.61·31-s + 1.39·33-s + 0.676·35-s + 0.986·37-s + 0.800·39-s − 1.24·41-s + 0.914·43-s + 0.298·45-s − 1.31·47-s + 6/7·49-s − 0.280·51-s − 0.412·53-s + 2.15·55-s + 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29593600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29593600\)    =    \(2^{12} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(1886.91\)
Root analytic conductor: \(6.59079\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 29593600,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
17$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) 2.3.b_c
7$D_{4}$ \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ac
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.11.i_bm
13$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.13.f_bc
19$D_{4}$ \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_bi
23$D_{4}$ \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_be
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \) 2.29.b_u
31$D_{4}$ \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.31.aj_da
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_co
41$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_be
43$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_da
47$D_{4}$ \( 1 + 9 T + 110 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.47.j_eg
53$D_{4}$ \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_ea
59$D_{4}$ \( 1 + 13 T + 122 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.59.n_es
61$D_{4}$ \( 1 + 19 T + 208 T^{2} + 19 p T^{3} + p^{2} T^{4} \) 2.61.t_ia
67$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_fm
71$D_{4}$ \( 1 - 21 T + 214 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.71.av_ig
73$D_{4}$ \( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.73.ah_ca
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_dy
89$D_{4}$ \( 1 + 7 T + 152 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.89.h_fw
97$D_{4}$ \( 1 - 9 T + 176 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_gu
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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

−7.915818215640767738140639144078, −7.56907044341038213506621207088, −7.51261872699765058931677440007, −6.96444181398855444226179090881, −6.44035850333500238324147369640, −6.30154912927293688576293339952, −5.80037678574700767424860896140, −5.37128125423571993484051531186, −4.99377789945894280707155255739, −4.88300652383944695049138906403, −4.42274679002303989733615722743, −3.97640421853068490492344353127, −3.21920276248370840318315189279, −3.07901965739458561986254309331, −2.66625642731110859057086656108, −2.39948126190682557041498779706, −1.59550704880028086857708408658, −0.73802318591659891450630980638, 0, 0, 0.73802318591659891450630980638, 1.59550704880028086857708408658, 2.39948126190682557041498779706, 2.66625642731110859057086656108, 3.07901965739458561986254309331, 3.21920276248370840318315189279, 3.97640421853068490492344353127, 4.42274679002303989733615722743, 4.88300652383944695049138906403, 4.99377789945894280707155255739, 5.37128125423571993484051531186, 5.80037678574700767424860896140, 6.30154912927293688576293339952, 6.44035850333500238324147369640, 6.96444181398855444226179090881, 7.51261872699765058931677440007, 7.56907044341038213506621207088, 7.915818215640767738140639144078

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