Properties

Label 4-429e2-1.1-c1e2-0-5
Degree $4$
Conductor $184041$
Sign $-1$
Analytic cond. $11.7346$
Root an. cond. $1.85083$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + 9-s − 4·11-s − 3·16-s − 8·23-s − 6·25-s − 4·31-s + 36-s + 4·37-s − 4·44-s + 24·47-s + 10·49-s − 20·53-s − 16·59-s − 7·64-s + 4·67-s + 81-s − 16·89-s − 8·92-s + 12·97-s − 4·99-s − 6·100-s − 24·103-s − 20·113-s + 5·121-s − 4·124-s + ⋯
L(s)  = 1  + 1/2·4-s + 1/3·9-s − 1.20·11-s − 3/4·16-s − 1.66·23-s − 6/5·25-s − 0.718·31-s + 1/6·36-s + 0.657·37-s − 0.603·44-s + 3.50·47-s + 10/7·49-s − 2.74·53-s − 2.08·59-s − 7/8·64-s + 0.488·67-s + 1/9·81-s − 1.69·89-s − 0.834·92-s + 1.21·97-s − 0.402·99-s − 3/5·100-s − 2.36·103-s − 1.88·113-s + 5/11·121-s − 0.359·124-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: yes
Self-dual: yes
Analytic rank: \(1\)
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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.2.a_ab
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.i_bu
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.29.a_ac
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.e_be
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.41.a_ac
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.47.ay_je
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.53.u_hy
59$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.q_gk
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ae_dy
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.79.a_adm
83$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.83.a_abq
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.q_je
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.am_gk
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.031161491518473380577105875280, −8.297569459397323685292333681389, −7.77174369691299465310382005001, −7.60683760460774971827660623785, −7.10385526543997740306176921337, −6.38669838330552276886820696719, −5.92498989233777985317497187718, −5.58950435267556785163234976003, −4.85239192007484847409189860881, −4.17694278689320689537342124933, −3.85394699418233027244613940996, −2.77795421892696621249439311331, −2.37919440715793867851950322806, −1.62341592053942670327770451781, 0, 1.62341592053942670327770451781, 2.37919440715793867851950322806, 2.77795421892696621249439311331, 3.85394699418233027244613940996, 4.17694278689320689537342124933, 4.85239192007484847409189860881, 5.58950435267556785163234976003, 5.92498989233777985317497187718, 6.38669838330552276886820696719, 7.10385526543997740306176921337, 7.60683760460774971827660623785, 7.77174369691299465310382005001, 8.297569459397323685292333681389, 9.031161491518473380577105875280

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