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Properties

Label 4-462e2-1.1-c1e2-0-11
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $13.6093$
Root an. cond. $1.92070$
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 − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s + 4·13-s − 4·14-s + 5·16-s − 6·18-s + 4·19-s + 4·21-s − 4·22-s − 8·24-s + 2·25-s − 8·26-s + 4·27-s + 6·28-s − 12·29-s + 4·31-s − 6·32-s + 4·33-s + 9·36-s + 4·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s + 1.10·13-s − 1.06·14-s + 5/4·16-s − 1.41·18-s + 0.917·19-s + 0.872·21-s − 0.852·22-s − 1.63·24-s + 2/5·25-s − 1.56·26-s + 0.769·27-s + 1.13·28-s − 2.22·29-s + 0.718·31-s − 1.06·32-s + 0.696·33-s + 3/2·36-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(13.6093\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.762406373\)
\(L(\frac12)\) \(\approx\) \(1.762406373\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_be
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_cc
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_be
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_cc
47$D_{4}$ \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.47.am_eo
53$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_dq
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.67.aq_fu
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$D_{4}$ \( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_fu
79$D_{4}$ \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.79.i_ew
83$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_dq
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_gk
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.17998808305582310956809182042, −10.74433393642895475367766421449, −10.07863628872376377350248640291, −10.00110105167259080413415084245, −9.214714422376758870179865567047, −9.032638248930660592329284262401, −8.617718601682383826649694374982, −8.317512233996301506091531949146, −7.58015131040833785076560168365, −7.53358436179697876158945181235, −6.93845055513310391012329955760, −6.39419680149113395802921236634, −5.67761341162773234140283236806, −5.28117544514644935282791954773, −4.11932092109000370232963063510, −3.90942912380034135356675869307, −3.07281231850540375016606100273, −2.46076100486846570220703570998, −1.62497531587841803918805722140, −1.12602938014071185532476208364, 1.12602938014071185532476208364, 1.62497531587841803918805722140, 2.46076100486846570220703570998, 3.07281231850540375016606100273, 3.90942912380034135356675869307, 4.11932092109000370232963063510, 5.28117544514644935282791954773, 5.67761341162773234140283236806, 6.39419680149113395802921236634, 6.93845055513310391012329955760, 7.53358436179697876158945181235, 7.58015131040833785076560168365, 8.317512233996301506091531949146, 8.617718601682383826649694374982, 9.032638248930660592329284262401, 9.214714422376758870179865567047, 10.00110105167259080413415084245, 10.07863628872376377350248640291, 10.74433393642895475367766421449, 11.17998808305582310956809182042

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