Properties

Label 4-74e2-1.1-c1e2-0-0
Degree $4$
Conductor $5476$
Sign $1$
Analytic cond. $0.349154$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 5-s − 2·6-s − 8-s + 3·9-s + 10-s + 4·11-s + 6·13-s − 2·15-s − 16-s − 3·17-s + 3·18-s − 2·19-s + 4·22-s − 12·23-s + 2·24-s + 5·25-s + 6·26-s − 10·27-s − 18·29-s − 2·30-s + 20·31-s − 8·33-s − 3·34-s − 37-s − 2·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.447·5-s − 0.816·6-s − 0.353·8-s + 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s − 0.516·15-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.458·19-s + 0.852·22-s − 2.50·23-s + 0.408·24-s + 25-s + 1.17·26-s − 1.92·27-s − 3.34·29-s − 0.365·30-s + 3.59·31-s − 1.39·33-s − 0.514·34-s − 0.164·37-s − 0.324·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5476\)    =    \(2^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(0.349154\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5476,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9137592447\)
\(L(\frac12)\) \(\approx\) \(0.9137592447\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
37$C_2$ \( 1 + T + p T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2^2$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) 2.5.ab_ae
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.17.d_ai
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.29.s_fj
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.43.q_fu
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.47.ae_du
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.53.ag_ar
59$C_2^2$ \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_f
61$C_2^2$ \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.61.af_abk
67$C_2^2$ \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_abf
71$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.71.a_act
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2^2$ \( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.79.g_abr
83$C_2^2$ \( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.83.c_adb
89$C_2^2$ \( 1 - 13 T + 80 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.89.an_dc
97$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.97.ag_hv
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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

−15.17909447899250491024396027080, −13.91701948656828477372271291806, −13.83401844787887754104842699083, −13.23250139058814471947182016311, −12.82618908656418296047681626106, −11.95528096197997345111936167598, −11.50907661287921777841709860176, −11.45836282369062742200235662188, −10.36792747105730767268102313947, −10.09457197427903449832368084192, −9.234038368076750619626841887352, −8.659543060140600794180498071726, −7.897310048569755711189734443433, −6.79701668017853675435396047183, −6.11612731662786134262438330044, −6.11451485117059988995423096877, −5.14059977250709566552519173371, −4.08846483113855139398748242624, −3.81416583544449140217727258991, −1.81063352494896351087694676103, 1.81063352494896351087694676103, 3.81416583544449140217727258991, 4.08846483113855139398748242624, 5.14059977250709566552519173371, 6.11451485117059988995423096877, 6.11612731662786134262438330044, 6.79701668017853675435396047183, 7.897310048569755711189734443433, 8.659543060140600794180498071726, 9.234038368076750619626841887352, 10.09457197427903449832368084192, 10.36792747105730767268102313947, 11.45836282369062742200235662188, 11.50907661287921777841709860176, 11.95528096197997345111936167598, 12.82618908656418296047681626106, 13.23250139058814471947182016311, 13.83401844787887754104842699083, 13.91701948656828477372271291806, 15.17909447899250491024396027080

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