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Properties

Label 4-2667e2-1.1-c1e2-0-1
Degree $4$
Conductor $7112889$
Sign $1$
Analytic cond. $453.523$
Root an. cond. $4.61477$
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  + 2·3-s + 2·4-s + 2·7-s + 3·9-s + 4·12-s + 4·13-s − 6·17-s + 4·19-s + 4·21-s + 12·23-s − 4·25-s + 4·27-s + 4·28-s − 6·29-s + 16·31-s + 6·36-s − 14·37-s + 8·39-s + 6·41-s + 4·43-s − 24·47-s + 3·49-s − 12·51-s + 8·52-s − 6·53-s + 8·57-s + 4·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 4-s + 0.755·7-s + 9-s + 1.15·12-s + 1.10·13-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 2.50·23-s − 4/5·25-s + 0.769·27-s + 0.755·28-s − 1.11·29-s + 2.87·31-s + 36-s − 2.30·37-s + 1.28·39-s + 0.937·41-s + 0.609·43-s − 3.50·47-s + 3/7·49-s − 1.68·51-s + 1.10·52-s − 0.824·53-s + 1.05·57-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7112889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7112889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7112889\)    =    \(3^{2} \cdot 7^{2} \cdot 127^{2}\)
Sign: $1$
Analytic conductor: \(453.523\)
Root analytic conductor: \(4.61477\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7112889,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.026467746\)
\(L(\frac12)\) \(\approx\) \(7.026467746\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
127$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_g
17$D_{4}$ \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_bl
19$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bk
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$D_{4}$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_br
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.37.o_et
41$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.41.ag_dh
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$D_{4}$ \( 1 + 6 T + 91 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.53.g_dn
59$C_2^2$ \( 1 + 112 T^{2} + p^{2} T^{4} \) 2.59.a_ei
61$D_{4}$ \( 1 - 4 T + 120 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_eq
67$D_{4}$ \( 1 - 4 T - 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_am
71$D_{4}$ \( 1 - 12 T + 172 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gq
73$D_{4}$ \( 1 + 8 T + 108 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_ee
79$D_{4}$ \( 1 + 2 T + 135 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_ff
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.89.a_de
97$D_{4}$ \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.97.ak_gj
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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

−9.002106697574846549238966804383, −8.582961843504661814972934814529, −8.333854013153734751896655491834, −7.82163518079327196831331359507, −7.74851595444853039329574339737, −7.00771028189804565477611062757, −6.88236497167738781734271983423, −6.43796658552937101059481167412, −6.29492396907212322616902875580, −5.45383735625270982849180921939, −4.98718079212420576325213364252, −4.82794472560227298101268428262, −4.19326088927695085362579266290, −3.76272811892513590923826996963, −3.13258262158209135687308174415, −2.97568520213590850281150463075, −2.43696633240949063460412632614, −1.68441610522505223825387644787, −1.66170372175561095069064860899, −0.796797708794726098704515066891, 0.796797708794726098704515066891, 1.66170372175561095069064860899, 1.68441610522505223825387644787, 2.43696633240949063460412632614, 2.97568520213590850281150463075, 3.13258262158209135687308174415, 3.76272811892513590923826996963, 4.19326088927695085362579266290, 4.82794472560227298101268428262, 4.98718079212420576325213364252, 5.45383735625270982849180921939, 6.29492396907212322616902875580, 6.43796658552937101059481167412, 6.88236497167738781734271983423, 7.00771028189804565477611062757, 7.74851595444853039329574339737, 7.82163518079327196831331359507, 8.333854013153734751896655491834, 8.582961843504661814972934814529, 9.002106697574846549238966804383

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