Properties

Label 4-429e2-1.1-c1e2-0-8
Degree $4$
Conductor $184041$
Sign $1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
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-s − 2·5-s − 4·7-s + 3·9-s − 2·11-s + 2·12-s − 2·13-s + 4·15-s − 3·16-s + 10·17-s − 8·19-s + 2·20-s + 8·21-s − 4·23-s − 4·25-s − 4·27-s + 4·28-s + 2·29-s − 6·31-s + 4·33-s + 8·35-s − 3·36-s − 4·37-s + 4·39-s − 4·41-s − 2·43-s + ⋯
L(s)  = 1  − 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 9-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 1.03·15-s − 3/4·16-s + 2.42·17-s − 1.83·19-s + 0.447·20-s + 1.74·21-s − 0.834·23-s − 4/5·25-s − 0.769·27-s + 0.755·28-s + 0.371·29-s − 1.07·31-s + 0.696·33-s + 1.35·35-s − 1/2·36-s − 0.657·37-s + 0.640·39-s − 0.624·41-s − 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(184041\)    =    \(3^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(11.7346\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 184041,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$D_{4}$ \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.5.c_i
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.7.e_s
17$D_{4}$ \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.17.ak_ce
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.19.i_bq
23$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.23.e_by
29$D_{4}$ \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.29.ac_bg
31$D_{4}$ \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_bs
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cw
43$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_ci
47$D_{4}$ \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_di
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$D_{4}$ \( 1 + 20 T + 206 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_hy
61$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_da
67$D_{4}$ \( 1 + 14 T + 156 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.67.o_ga
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \) 2.71.a_dq
73$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_ek
79$D_{4}$ \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_ee
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.83.a_aba
89$D_{4}$ \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.89.o_fw
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.97.ae_fu
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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

−10.98233132089374916908110440536, −10.31850574761818891989063494270, −10.09122356878340641242633577971, −9.887918448489030988404690435313, −9.140383791686041291828294967045, −8.810786877525986769989149208569, −7.943656972539248090553778634377, −7.73888656519421108670134539450, −7.24340841132030452266532747571, −6.62780667496321996725889291267, −6.08334615241733373529407954872, −5.85709950063880528463459733171, −5.11423171313381413795979310471, −4.64213184103502313720127885293, −3.97710819556428815156256206048, −3.55474922112396448198929899788, −2.87032313039725980496591990680, −1.73227011454809870487556010975, 0, 0, 1.73227011454809870487556010975, 2.87032313039725980496591990680, 3.55474922112396448198929899788, 3.97710819556428815156256206048, 4.64213184103502313720127885293, 5.11423171313381413795979310471, 5.85709950063880528463459733171, 6.08334615241733373529407954872, 6.62780667496321996725889291267, 7.24340841132030452266532747571, 7.73888656519421108670134539450, 7.943656972539248090553778634377, 8.810786877525986769989149208569, 9.140383791686041291828294967045, 9.887918448489030988404690435313, 10.09122356878340641242633577971, 10.31850574761818891989063494270, 10.98233132089374916908110440536

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