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Properties

Label 4-62e2-1.1-c1e2-0-2
Degree $4$
Conductor $3844$
Sign $1$
Analytic cond. $0.245096$
Root an. cond. $0.703613$
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 − 3·3-s + 3·4-s − 5-s − 6·6-s + 3·7-s + 4·8-s + 3·9-s − 2·10-s + 3·11-s − 9·12-s − 5·13-s + 6·14-s + 3·15-s + 5·16-s − 3·17-s + 6·18-s − 7·19-s − 3·20-s − 9·21-s + 6·22-s − 8·23-s − 12·24-s + 5·25-s − 10·26-s + 9·28-s + 4·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.73·3-s + 3/2·4-s − 0.447·5-s − 2.44·6-s + 1.13·7-s + 1.41·8-s + 9-s − 0.632·10-s + 0.904·11-s − 2.59·12-s − 1.38·13-s + 1.60·14-s + 0.774·15-s + 5/4·16-s − 0.727·17-s + 1.41·18-s − 1.60·19-s − 0.670·20-s − 1.96·21-s + 1.27·22-s − 1.66·23-s − 2.44·24-s + 25-s − 1.96·26-s + 1.70·28-s + 0.742·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3844\)    =    \(2^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(0.245096\)
Root analytic conductor: \(0.703613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3844,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.010608080\)
\(L(\frac12)\) \(\approx\) \(1.010608080\)
\(L(\frac{3}{2})\) 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)$
bad2$C_1$ \( ( 1 - T )^{2} \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 5 T - 58 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
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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

−15.07215087684373906727504353545, −14.54671575955381139397634967053, −14.49586798721203059868545651464, −13.69140718218774055013063980316, −12.73513252130305159402070699571, −12.51084467933259813764398310948, −11.76314149804396347852035739965, −11.64340725718200534139312562372, −11.18846220196795281328918238964, −10.54210230380839543941245377517, −9.998128682271651124720912425896, −8.718211811453499604428634161231, −8.001071386375624658516722625264, −7.15663822429982815627228701898, −6.36717267283488216493841864950, −6.10879142913240382362487661533, −5.01827056275199137447909471844, −4.69882611932801093416359433427, −4.03457168001769038398374929217, −2.26389338676405307833668279124, 2.26389338676405307833668279124, 4.03457168001769038398374929217, 4.69882611932801093416359433427, 5.01827056275199137447909471844, 6.10879142913240382362487661533, 6.36717267283488216493841864950, 7.15663822429982815627228701898, 8.001071386375624658516722625264, 8.718211811453499604428634161231, 9.998128682271651124720912425896, 10.54210230380839543941245377517, 11.18846220196795281328918238964, 11.64340725718200534139312562372, 11.76314149804396347852035739965, 12.51084467933259813764398310948, 12.73513252130305159402070699571, 13.69140718218774055013063980316, 14.49586798721203059868545651464, 14.54671575955381139397634967053, 15.07215087684373906727504353545

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