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Properties

Label 4-474e2-1.1-c1e2-0-8
Degree $4$
Conductor $224676$
Sign $1$
Analytic cond. $14.3255$
Root an. cond. $1.94548$
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  − 2·2-s + 2·3-s + 3·4-s + 3·5-s − 4·6-s − 7-s − 4·8-s + 3·9-s − 6·10-s + 8·11-s + 6·12-s − 4·13-s + 2·14-s + 6·15-s + 5·16-s + 7·17-s − 6·18-s + 3·19-s + 9·20-s − 2·21-s − 16·22-s − 3·23-s − 8·24-s − 2·25-s + 8·26-s + 4·27-s − 3·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.34·5-s − 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s − 1.89·10-s + 2.41·11-s + 1.73·12-s − 1.10·13-s + 0.534·14-s + 1.54·15-s + 5/4·16-s + 1.69·17-s − 1.41·18-s + 0.688·19-s + 2.01·20-s − 0.436·21-s − 3.41·22-s − 0.625·23-s − 1.63·24-s − 2/5·25-s + 1.56·26-s + 0.769·27-s − 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(224676\)    =    \(2^{2} \cdot 3^{2} \cdot 79^{2}\)
Sign: $1$
Analytic conductor: \(14.3255\)
Root analytic conductor: \(1.94548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 224676,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.064172298\)
\(L(\frac12)\) \(\approx\) \(2.064172298\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
79$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.5.ad_l
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_d
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.11.ai_bm
13$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_k
17$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.17.ah_bt
19$D_{4}$ \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.19.ad_j
23$D_{4}$ \( 1 + 3 T - 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_an
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.31.e_co
37$D_{4}$ \( 1 + 15 T + 119 T^{2} + 15 p T^{3} + p^{2} T^{4} \) 2.37.p_ep
41$D_{4}$ \( 1 - 9 T + 101 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.41.aj_dx
43$D_{4}$ \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_cs
47$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.47.ao_du
53$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_fu
59$D_{4}$ \( 1 - 7 T + 119 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.59.ah_ep
61$D_{4}$ \( 1 + 5 T + 127 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_ex
67$D_{4}$ \( 1 - 7 T + 115 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.67.ah_el
71$D_{4}$ \( 1 - 8 T + 138 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.71.ai_fi
73$D_{4}$ \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.73.ad_ad
83$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_bq
89$D_{4}$ \( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.89.u_jy
97$D_{4}$ \( 1 + 23 T + 265 T^{2} + 23 p T^{3} + p^{2} T^{4} \) 2.97.x_kf
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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

−10.79855591282060024135867046424, −10.70503882606646309048532543860, −9.824253101612906808156283928539, −9.704888983702881455339429841675, −9.436739170913763447114390863273, −9.400791082943993219071083438215, −8.552339870491815710377472073857, −8.324356648079937786570993805379, −7.60838575451028402935056748719, −7.30754530310198964491270429450, −6.65044191522850645230534713667, −6.51388771040525126300430480343, −5.61560127952099103356191236968, −5.45200301721145153843731622501, −4.18607246131466104745063402329, −3.73209017142579808778547812614, −3.00980001229884436035175696592, −2.41716878250729208280964921350, −1.62097244344141021510230345690, −1.23859641607495351356289776034, 1.23859641607495351356289776034, 1.62097244344141021510230345690, 2.41716878250729208280964921350, 3.00980001229884436035175696592, 3.73209017142579808778547812614, 4.18607246131466104745063402329, 5.45200301721145153843731622501, 5.61560127952099103356191236968, 6.51388771040525126300430480343, 6.65044191522850645230534713667, 7.30754530310198964491270429450, 7.60838575451028402935056748719, 8.324356648079937786570993805379, 8.552339870491815710377472073857, 9.400791082943993219071083438215, 9.436739170913763447114390863273, 9.704888983702881455339429841675, 9.824253101612906808156283928539, 10.70503882606646309048532543860, 10.79855591282060024135867046424

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