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Properties

Label 4-1430e2-1.1-c1e2-0-3
Degree $4$
Conductor $2044900$
Sign $-1$
Analytic cond. $130.384$
Root an. cond. $3.37914$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s + 4-s + 2·5-s + 6·9-s − 6·11-s − 4·12-s − 8·15-s + 16-s + 2·20-s + 12·23-s + 3·25-s + 4·27-s + 4·31-s + 24·33-s + 6·36-s + 4·37-s − 6·44-s + 12·45-s − 24·47-s − 4·48-s + 2·49-s + 12·53-s − 12·55-s + 12·59-s − 8·60-s + 64-s − 8·67-s + ⋯
L(s)  = 1  − 2.30·3-s + 1/2·4-s + 0.894·5-s + 2·9-s − 1.80·11-s − 1.15·12-s − 2.06·15-s + 1/4·16-s + 0.447·20-s + 2.50·23-s + 3/5·25-s + 0.769·27-s + 0.718·31-s + 4.17·33-s + 36-s + 0.657·37-s − 0.904·44-s + 1.78·45-s − 3.50·47-s − 0.577·48-s + 2/7·49-s + 1.64·53-s − 1.61·55-s + 1.56·59-s − 1.03·60-s + 1/8·64-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2044900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2044900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2044900\)    =    \(2^{2} \cdot 5^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(130.384\)
Root analytic conductor: \(3.37914\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2044900,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.23.am_de
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.43.a_de
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.67.i_fu
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
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

−7.13284046687725717765533260434, −7.01938464652870881402898826049, −6.55827234630368931511436644185, −6.21865412034335443989351794245, −5.70337012148360379248131749579, −5.38356125373378726827971266003, −5.20912025979554566633790456489, −4.79474315687223740281257481843, −4.37051942916216409512438466875, −3.22135578238067272752399472859, −2.83582044678427759968275472285, −2.48318029352373914558980684814, −1.46367458730915917590656952546, −0.870890047860011749751230498814, 0, 0.870890047860011749751230498814, 1.46367458730915917590656952546, 2.48318029352373914558980684814, 2.83582044678427759968275472285, 3.22135578238067272752399472859, 4.37051942916216409512438466875, 4.79474315687223740281257481843, 5.20912025979554566633790456489, 5.38356125373378726827971266003, 5.70337012148360379248131749579, 6.21865412034335443989351794245, 6.55827234630368931511436644185, 7.01938464652870881402898826049, 7.13284046687725717765533260434

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