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Properties

Label 4-8512e2-1.1-c1e2-0-22
Degree $4$
Conductor $72454144$
Sign $1$
Analytic cond. $4619.73$
Root an. cond. $8.24431$
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  − 3·3-s − 2·5-s + 2·7-s + 2·9-s + 11-s + 2·13-s + 6·15-s + 17-s + 2·19-s − 6·21-s − 2·23-s − 7·25-s + 6·27-s − 5·29-s − 31-s − 3·33-s − 4·35-s + 14·37-s − 6·39-s + 9·41-s − 8·43-s − 4·45-s + 6·47-s + 3·49-s − 3·51-s − 3·53-s − 2·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.894·5-s + 0.755·7-s + 2/3·9-s + 0.301·11-s + 0.554·13-s + 1.54·15-s + 0.242·17-s + 0.458·19-s − 1.30·21-s − 0.417·23-s − 7/5·25-s + 1.15·27-s − 0.928·29-s − 0.179·31-s − 0.522·33-s − 0.676·35-s + 2.30·37-s − 0.960·39-s + 1.40·41-s − 1.21·43-s − 0.596·45-s + 0.875·47-s + 3/7·49-s − 0.420·51-s − 0.412·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72454144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72454144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(72454144\)    =    \(2^{12} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(4619.73\)
Root analytic conductor: \(8.24431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 72454144,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_h
5$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.5.c_l
11$C_4$ \( 1 - T + 21 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_v
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$D_{4}$ \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) 2.17.ab_x
23$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_bb
29$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.29.f_cl
31$C_4$ \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) 2.31.b_abn
37$D_{4}$ \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.37.ao_dz
41$C_4$ \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_ct
43$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_de
47$D_{4}$ \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_df
53$D_{4}$ \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_dt
59$D_{4}$ \( 1 + 20 T + 213 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_if
61$D_{4}$ \( 1 - 6 T + 51 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_bz
67$D_{4}$ \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.67.l_dz
71$D_{4}$ \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_dx
73$D_{4}$ \( 1 + 7 T + 97 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.73.h_dt
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \) 2.79.a_fi
83$D_{4}$ \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) 2.83.n_hp
89$D_{4}$ \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.89.ak_hq
97$D_{4}$ \( 1 - 6 T + 183 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.97.ag_hb
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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

−7.64756280701238665251816556217, −7.47308702478798969273996422072, −6.68788641542302881211882749618, −6.62001129913886742269465180081, −5.95520742058219496565112847723, −5.93898333016967380782777476436, −5.55508661559895860874553987720, −5.47050394873346192930961020813, −4.65176543000716991258913729716, −4.54762714835190285692115357424, −4.24130790243383645241274678958, −3.86131914304230977506614567051, −3.21971756233377201205985260427, −3.12120225284675902035291227604, −2.26627808239898412619332172064, −1.95591765777815027522453746096, −1.13898682806901433992598212277, −1.01247274694987399627713147355, 0, 0, 1.01247274694987399627713147355, 1.13898682806901433992598212277, 1.95591765777815027522453746096, 2.26627808239898412619332172064, 3.12120225284675902035291227604, 3.21971756233377201205985260427, 3.86131914304230977506614567051, 4.24130790243383645241274678958, 4.54762714835190285692115357424, 4.65176543000716991258913729716, 5.47050394873346192930961020813, 5.55508661559895860874553987720, 5.93898333016967380782777476436, 5.95520742058219496565112847723, 6.62001129913886742269465180081, 6.68788641542302881211882749618, 7.47308702478798969273996422072, 7.64756280701238665251816556217

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