Properties

Label 4-3648e2-1.1-c1e2-0-16
Degree $4$
Conductor $13307904$
Sign $1$
Analytic cond. $848.523$
Root an. cond. $5.39716$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s + 3·7-s + 3·9-s + 3·11-s − 12·13-s − 2·15-s + 17-s + 2·19-s − 6·21-s − 8·23-s + 25-s − 4·27-s − 4·29-s − 6·31-s − 6·33-s + 3·35-s − 6·37-s + 24·39-s + 6·41-s + 43-s + 3·45-s − 5·47-s + 3·49-s − 2·51-s + 12·53-s + 3·55-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s + 1.13·7-s + 9-s + 0.904·11-s − 3.32·13-s − 0.516·15-s + 0.242·17-s + 0.458·19-s − 1.30·21-s − 1.66·23-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 1.07·31-s − 1.04·33-s + 0.507·35-s − 0.986·37-s + 3.84·39-s + 0.937·41-s + 0.152·43-s + 0.447·45-s − 0.729·47-s + 3/7·49-s − 0.280·51-s + 1.64·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13307904\)    =    \(2^{12} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(848.523\)
Root analytic conductor: \(5.39716\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 13307904,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$D_{4}$ \( 1 - T - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 90 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 9 T + 132 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$D_{4}$ \( 1 - 7 T + 148 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−8.077342206820065567765041244831, −7.943507762412992174271767049092, −7.47786635663804184319580446091, −7.27658138751254972395870708021, −6.95282275221969544631165743461, −6.55175999568645575358227596781, −5.84839698076711898294095629094, −5.79027959541990965954216541789, −5.30652218538457381897603655384, −4.99835657264466679272277262530, −4.65401599809122212921915955933, −4.28812977283282199851529238053, −3.90193039295762445305134641944, −3.24659206653404389613743240261, −2.42865711470263501591328849768, −2.31284430677716258586350443212, −1.53915144002156966880209219833, −1.37273261810207820816765784721, 0, 0, 1.37273261810207820816765784721, 1.53915144002156966880209219833, 2.31284430677716258586350443212, 2.42865711470263501591328849768, 3.24659206653404389613743240261, 3.90193039295762445305134641944, 4.28812977283282199851529238053, 4.65401599809122212921915955933, 4.99835657264466679272277262530, 5.30652218538457381897603655384, 5.79027959541990965954216541789, 5.84839698076711898294095629094, 6.55175999568645575358227596781, 6.95282275221969544631165743461, 7.27658138751254972395870708021, 7.47786635663804184319580446091, 7.943507762412992174271767049092, 8.077342206820065567765041244831

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