Properties

Label 4-494e2-1.1-c1e2-0-2
Degree $4$
Conductor $244036$
Sign $1$
Analytic cond. $15.5599$
Root an. cond. $1.98610$
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 − 2·3-s − 6·5-s − 2·6-s − 8-s + 3·9-s − 6·10-s + 6·11-s + 7·13-s + 12·15-s − 16-s − 7·17-s + 3·18-s − 19-s + 6·22-s + 4·23-s + 2·24-s + 17·25-s + 7·26-s − 10·27-s + 9·29-s + 12·30-s − 12·33-s − 7·34-s − 3·37-s − 38-s − 14·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 2.68·5-s − 0.816·6-s − 0.353·8-s + 9-s − 1.89·10-s + 1.80·11-s + 1.94·13-s + 3.09·15-s − 1/4·16-s − 1.69·17-s + 0.707·18-s − 0.229·19-s + 1.27·22-s + 0.834·23-s + 0.408·24-s + 17/5·25-s + 1.37·26-s − 1.92·27-s + 1.67·29-s + 2.19·30-s − 2.08·33-s − 1.20·34-s − 0.493·37-s − 0.162·38-s − 2.24·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(244036\)    =    \(2^{2} \cdot 13^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(15.5599\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 244036,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8068947080\)
\(L(\frac12)\) \(\approx\) \(0.8068947080\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
13$C_2$ \( 1 - 7 T + p T^{2} \)
19$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.3.c_b
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_z
17$C_2^2$ \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.17.h_bg
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.29.aj_ca
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.37.d_abc
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.41.f_aq
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.43.a_abr
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.53.aw_it
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_bp
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.n_ee
67$C_2^2$ \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.67.k_bh
71$C_2^2$ \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_abj
73$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.73.c_fr
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2^2$ \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_adh
97$C_2^2$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.97.s_it
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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

−11.49326727403646771760566642809, −11.09747864870647920183559005434, −10.54620790397148903272851159571, −10.12994967066220652488011789481, −9.018222222981780651762541338373, −8.969777733314537011666109514976, −8.378608277114755594076904143272, −8.211513254541930281601660337113, −7.17267975264973778176562236624, −6.87053308679075773416660422953, −6.85064927496204764752480267123, −5.82975582141793523864290918747, −5.80767633142951800554943572963, −4.57753541578017524327109030925, −4.38792335838834999003029745459, −4.04962745635648292571403733604, −3.69739378735013550963920455381, −3.05818779208358888201401565589, −1.50505340427546426357281455683, −0.57005678907691320706357970845, 0.57005678907691320706357970845, 1.50505340427546426357281455683, 3.05818779208358888201401565589, 3.69739378735013550963920455381, 4.04962745635648292571403733604, 4.38792335838834999003029745459, 4.57753541578017524327109030925, 5.80767633142951800554943572963, 5.82975582141793523864290918747, 6.85064927496204764752480267123, 6.87053308679075773416660422953, 7.17267975264973778176562236624, 8.211513254541930281601660337113, 8.378608277114755594076904143272, 8.969777733314537011666109514976, 9.018222222981780651762541338373, 10.12994967066220652488011789481, 10.54620790397148903272851159571, 11.09747864870647920183559005434, 11.49326727403646771760566642809

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