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Properties

Label 4-1197e2-1.1-c1e2-0-5
Degree $4$
Conductor $1432809$
Sign $1$
Analytic cond. $91.3571$
Root an. cond. $3.09161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + 2·5-s + 7-s − 3·11-s + 4·13-s − 7·17-s + 19-s + 4·20-s + 5·23-s + 5·25-s + 2·28-s − 4·29-s − 10·31-s + 2·35-s − 8·37-s − 12·41-s + 24·43-s − 6·44-s − 5·47-s − 6·49-s + 8·52-s + 4·53-s − 6·55-s + 14·59-s + 13·61-s − 8·64-s + 8·65-s + ⋯
L(s)  = 1  + 4-s + 0.894·5-s + 0.377·7-s − 0.904·11-s + 1.10·13-s − 1.69·17-s + 0.229·19-s + 0.894·20-s + 1.04·23-s + 25-s + 0.377·28-s − 0.742·29-s − 1.79·31-s + 0.338·35-s − 1.31·37-s − 1.87·41-s + 3.65·43-s − 0.904·44-s − 0.729·47-s − 6/7·49-s + 1.10·52-s + 0.549·53-s − 0.809·55-s + 1.82·59-s + 1.66·61-s − 64-s + 0.992·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1432809 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1432809\)    =    \(3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(91.3571\)
Root analytic conductor: \(3.09161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1432809,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.318038152\)
\(L(\frac12)\) \(\approx\) \(3.318038152\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
19$C_2$ \( 1 - T + T^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.11.d_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bg
23$C_2^2$ \( 1 - 5 T + 2 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.23.af_c
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_cr
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.37.i_bb
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.43.ay_iw
47$C_2^2$ \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.47.f_aw
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_abl
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fh
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.61.an_ee
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.71.au_ji
73$C_2^2$ \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) 2.73.b_acu
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.83.as_jn
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_jb
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.97.am_iw
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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

−10.01475256370326815411411489735, −9.380402981580854172671331157514, −9.186089208171209927997694413526, −8.784198014147347797834613386619, −8.387954133710412312981801478666, −7.939792917997030024454190316784, −7.19834983444193913271334574946, −7.15794408352459252249858922352, −6.55986678383901380308617186732, −6.40269430681634937224962408065, −5.69253610454756017364130226273, −5.25021013409811545626589968394, −5.12836215565507625946555648022, −4.37708533545215457938505764914, −3.63407390337203064714297090050, −3.36949448772577536712305346600, −2.32661254643518417828854216703, −2.28271469726049107960112444562, −1.75372935083218540618336424563, −0.75970907950683873875276380234, 0.75970907950683873875276380234, 1.75372935083218540618336424563, 2.28271469726049107960112444562, 2.32661254643518417828854216703, 3.36949448772577536712305346600, 3.63407390337203064714297090050, 4.37708533545215457938505764914, 5.12836215565507625946555648022, 5.25021013409811545626589968394, 5.69253610454756017364130226273, 6.40269430681634937224962408065, 6.55986678383901380308617186732, 7.15794408352459252249858922352, 7.19834983444193913271334574946, 7.939792917997030024454190316784, 8.387954133710412312981801478666, 8.784198014147347797834613386619, 9.186089208171209927997694413526, 9.380402981580854172671331157514, 10.01475256370326815411411489735

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