Properties

Label 4-148e2-1.1-c1e2-0-2
Degree $4$
Conductor $21904$
Sign $1$
Analytic cond. $1.39661$
Root an. cond. $1.08709$
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  − 3-s + 3·5-s − 7-s + 3·9-s + 3·13-s − 3·15-s + 3·17-s − 3·19-s + 21-s + 25-s − 8·27-s − 3·35-s − 10·37-s − 3·39-s + 9·41-s + 9·45-s + 7·49-s − 3·51-s − 3·53-s + 3·57-s + 3·59-s + 3·61-s − 3·63-s + 9·65-s − 13·67-s + 15·71-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.377·7-s + 9-s + 0.832·13-s − 0.774·15-s + 0.727·17-s − 0.688·19-s + 0.218·21-s + 1/5·25-s − 1.53·27-s − 0.507·35-s − 1.64·37-s − 0.480·39-s + 1.40·41-s + 1.34·45-s + 49-s − 0.420·51-s − 0.412·53-s + 0.397·57-s + 0.390·59-s + 0.384·61-s − 0.377·63-s + 1.11·65-s − 1.58·67-s + 1.78·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21904\)    =    \(2^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(1.39661\)
Root analytic conductor: \(1.08709\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.253233154\)
\(L(\frac12)\) \(\approx\) \(1.253233154\)
\(L(\frac{3}{2})\) 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 \( 1 \)
37$C_2$ \( 1 + 10 T + p T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 3 T + 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−13.31926654546282965007327535817, −12.89136249982525335610123968547, −12.13414566041300272636046302071, −12.13327851989556179676455464939, −10.97941598612105839184187630383, −10.86760617101171680007706570030, −10.19114359251100614105433903405, −9.815861609170881495530350816701, −9.316529186390713359750239467917, −8.850072967066781618566475576878, −7.978225987983068392756754526568, −7.45567649746581603190623900529, −6.55606778270217966797590862193, −6.38868495218168073578554458417, −5.54149508142722165940545477844, −5.34632206423000692185500798038, −4.19992295178566342390624278530, −3.62510828017454753325735723436, −2.35823243719693822046831684597, −1.44115591648537293006599538106, 1.44115591648537293006599538106, 2.35823243719693822046831684597, 3.62510828017454753325735723436, 4.19992295178566342390624278530, 5.34632206423000692185500798038, 5.54149508142722165940545477844, 6.38868495218168073578554458417, 6.55606778270217966797590862193, 7.45567649746581603190623900529, 7.978225987983068392756754526568, 8.850072967066781618566475576878, 9.316529186390713359750239467917, 9.815861609170881495530350816701, 10.19114359251100614105433903405, 10.86760617101171680007706570030, 10.97941598612105839184187630383, 12.13327851989556179676455464939, 12.13414566041300272636046302071, 12.89136249982525335610123968547, 13.31926654546282965007327535817

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