Properties

Label 4-363e2-1.1-c1e2-0-16
Degree $4$
Conductor $131769$
Sign $-1$
Analytic cond. $8.40170$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·4-s + 4·7-s + 9-s − 6·12-s − 2·13-s + 5·16-s − 12·19-s + 8·21-s − 9·25-s − 4·27-s − 12·28-s − 4·31-s − 3·36-s − 6·37-s − 4·39-s + 10·48-s − 2·49-s + 6·52-s − 24·57-s − 12·61-s + 4·63-s − 3·64-s + 4·67-s + 4·73-s − 18·75-s + 36·76-s + ⋯
L(s)  = 1  + 1.15·3-s − 3/2·4-s + 1.51·7-s + 1/3·9-s − 1.73·12-s − 0.554·13-s + 5/4·16-s − 2.75·19-s + 1.74·21-s − 9/5·25-s − 0.769·27-s − 2.26·28-s − 0.718·31-s − 1/2·36-s − 0.986·37-s − 0.640·39-s + 1.44·48-s − 2/7·49-s + 0.832·52-s − 3.17·57-s − 1.53·61-s + 0.503·63-s − 3/8·64-s + 0.488·67-s + 0.468·73-s − 2.07·75-s + 4.12·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $-1$
Analytic conductor: \(8.40170\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 131769,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - 2 T + p T^{2} \)
11 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.5.a_j
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.17.a_j
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.19.m_cw
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.23.a_bq
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.41.a_cf
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.79.au_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.89.a_dt
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.97.ba_nz
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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

−9.127942331541421284163370204249, −8.507285126107606377633981082529, −8.264856530445176716222542589828, −8.010979443598278861073897866124, −7.49323810887805876503518242868, −6.74272981252089439582006678154, −5.97336865191641765106303088244, −5.44861874763839175210229504778, −4.68607726153727393582211790246, −4.50122243823954701224397495329, −3.89751875979123562820900427499, −3.33556810952068997685484956976, −2.02313919782203708817074154348, −1.98218366620434510274683932709, 0, 1.98218366620434510274683932709, 2.02313919782203708817074154348, 3.33556810952068997685484956976, 3.89751875979123562820900427499, 4.50122243823954701224397495329, 4.68607726153727393582211790246, 5.44861874763839175210229504778, 5.97336865191641765106303088244, 6.74272981252089439582006678154, 7.49323810887805876503518242868, 8.010979443598278861073897866124, 8.264856530445176716222542589828, 8.507285126107606377633981082529, 9.127942331541421284163370204249

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