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Properties

Label 4-14112-1.1-c1e2-0-2
Degree $4$
Conductor $14112$
Sign $1$
Analytic cond. $0.899793$
Root an. cond. $0.973947$
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-s + 4-s − 4·5-s + 8-s + 9-s − 4·10-s + 12·13-s + 16-s + 4·17-s + 18-s − 4·20-s + 2·25-s + 12·26-s − 4·29-s + 32-s + 4·34-s + 36-s − 20·37-s − 4·40-s − 12·41-s − 4·45-s + 49-s + 2·50-s + 12·52-s + 12·53-s − 4·58-s + 12·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.353·8-s + 1/3·9-s − 1.26·10-s + 3.32·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.894·20-s + 2/5·25-s + 2.35·26-s − 0.742·29-s + 0.176·32-s + 0.685·34-s + 1/6·36-s − 3.28·37-s − 0.632·40-s − 1.87·41-s − 0.596·45-s + 1/7·49-s + 0.282·50-s + 1.66·52-s + 1.64·53-s − 0.525·58-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14112 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14112\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.899793\)
Root analytic conductor: \(0.973947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14112,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.370183666\)
\(L(\frac12)\) \(\approx\) \(1.370183666\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.19.a_w
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.29.e_ck
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.a_eo
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−11.37318927677395242391128273111, −10.71603142151067115958484330836, −10.51951977961491945205840276286, −9.567727627901464767318320243097, −8.579197535720292899748335278003, −8.356918966632091922574869118017, −7.963562405403294064353399542187, −6.89499765960031010222506198709, −6.84552480602986516155667793601, −5.69761771888975023011122238972, −5.33985014787602837985094112170, −4.04752481141371955267217260485, −3.62482887081886485478101246558, −3.54052974946578322805924483948, −1.50800619952171259085588630937, 1.50800619952171259085588630937, 3.54052974946578322805924483948, 3.62482887081886485478101246558, 4.04752481141371955267217260485, 5.33985014787602837985094112170, 5.69761771888975023011122238972, 6.84552480602986516155667793601, 6.89499765960031010222506198709, 7.963562405403294064353399542187, 8.356918966632091922574869118017, 8.579197535720292899748335278003, 9.567727627901464767318320243097, 10.51951977961491945205840276286, 10.71603142151067115958484330836, 11.37318927677395242391128273111

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