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Properties

Label 4-1769472-1.1-c1e2-0-1
Degree $4$
Conductor $1769472$
Sign $1$
Analytic cond. $112.823$
Root an. cond. $3.25911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 9-s − 4·11-s − 6·17-s − 4·19-s + 2·25-s + 27-s − 4·33-s − 14·41-s + 6·49-s − 6·51-s − 4·57-s + 12·59-s + 16·67-s − 20·73-s + 2·75-s + 81-s − 12·83-s + 18·89-s + 16·97-s − 4·99-s − 20·107-s + 10·113-s + 6·121-s − 14·123-s + 127-s + 131-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.45·17-s − 0.917·19-s + 2/5·25-s + 0.192·27-s − 0.696·33-s − 2.18·41-s + 6/7·49-s − 0.840·51-s − 0.529·57-s + 1.56·59-s + 1.95·67-s − 2.34·73-s + 0.230·75-s + 1/9·81-s − 1.31·83-s + 1.90·89-s + 1.62·97-s − 0.402·99-s − 1.93·107-s + 0.940·113-s + 6/11·121-s − 1.26·123-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1769472 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1769472\)    =    \(2^{16} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(112.823\)
Root analytic conductor: \(3.25911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1769472,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.498032378\)
\(L(\frac12)\) \(\approx\) \(1.498032378\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.e_k
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.g_bi
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.e_bm
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.29.a_ac
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.37.a_s
41$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.o_es
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.47.a_ack
53$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.53.a_ak
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.61.a_de
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.aq_ha
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.u_iw
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.m_he
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.89.as_jy
97$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.aq_gc
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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

−7.84188103555278893187959189334, −7.43987353126491932373856450841, −6.95870060261989931162845703414, −6.58109062111164641439666363727, −6.30369937477715143348032874159, −5.47734093216756610769474863505, −5.24474947824441450139706033134, −4.72976679030323786993830080665, −4.21253039161606111071849899781, −3.83221383137522969437051195772, −3.14829738248562048511669896745, −2.63329532608116959549467292840, −2.19248080618599981935919748911, −1.66280979140269832745785849409, −0.46618205776326175064895064670, 0.46618205776326175064895064670, 1.66280979140269832745785849409, 2.19248080618599981935919748911, 2.63329532608116959549467292840, 3.14829738248562048511669896745, 3.83221383137522969437051195772, 4.21253039161606111071849899781, 4.72976679030323786993830080665, 5.24474947824441450139706033134, 5.47734093216756610769474863505, 6.30369937477715143348032874159, 6.58109062111164641439666363727, 6.95870060261989931162845703414, 7.43987353126491932373856450841, 7.84188103555278893187959189334

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