Properties

Label 4-1027e2-1.1-c1e2-0-0
Degree $4$
Conductor $1054729$
Sign $1$
Analytic cond. $67.2504$
Root an. cond. $2.86367$
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  − 4·3-s + 6·9-s − 6·13-s − 4·16-s + 4·17-s − 14·23-s + 6·25-s + 4·27-s − 20·29-s + 24·39-s + 8·43-s + 16·48-s + 13·49-s − 16·51-s + 4·53-s + 8·61-s + 56·69-s − 24·75-s − 2·79-s − 37·81-s + 80·87-s + 18·101-s − 28·103-s + 12·107-s − 36·117-s + 18·121-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s − 1.66·13-s − 16-s + 0.970·17-s − 2.91·23-s + 6/5·25-s + 0.769·27-s − 3.71·29-s + 3.84·39-s + 1.21·43-s + 2.30·48-s + 13/7·49-s − 2.24·51-s + 0.549·53-s + 1.02·61-s + 6.74·69-s − 2.77·75-s − 0.225·79-s − 4.11·81-s + 8.57·87-s + 1.79·101-s − 2.75·103-s + 1.16·107-s − 3.32·117-s + 1.63·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1054729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1054729\)    =    \(13^{2} \cdot 79^{2}\)
Sign: $1$
Analytic conductor: \(67.2504\)
Root analytic conductor: \(2.86367\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1054729,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad13$C_2$ \( 1 + 6 T + p T^{2} \)
79$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.23.o_dr
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.31.a_abu
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.41.a_acf
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \) 2.47.a_adp
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.59.a_ady
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.61.ai_fi
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.71.a_ada
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.89.a_aek
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−9.898996663303809312614236419983, −9.573509559051375311765496839442, −8.815872272707202900316199491501, −8.746101523040206346695550340297, −7.74507276659119902313056322636, −7.43373731358587861364137302336, −7.34069161003514321049633440147, −6.57830906601981968820965182191, −6.18492271226338655811132569860, −5.84025005742039968694678060435, −5.34382672279066834672417000702, −5.26003066661175038671313475196, −4.69430534667663482009719920267, −3.89131873852254113465427132131, −3.85156774784900775409628259078, −2.42537674649042609382712832217, −2.37611763235103715171176387613, −1.21805553966385498610696629093, 0, 0, 1.21805553966385498610696629093, 2.37611763235103715171176387613, 2.42537674649042609382712832217, 3.85156774784900775409628259078, 3.89131873852254113465427132131, 4.69430534667663482009719920267, 5.26003066661175038671313475196, 5.34382672279066834672417000702, 5.84025005742039968694678060435, 6.18492271226338655811132569860, 6.57830906601981968820965182191, 7.34069161003514321049633440147, 7.43373731358587861364137302336, 7.74507276659119902313056322636, 8.746101523040206346695550340297, 8.815872272707202900316199491501, 9.573509559051375311765496839442, 9.898996663303809312614236419983

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