Properties

Label 4-684e2-1.1-c1e2-0-14
Degree $4$
Conductor $467856$
Sign $1$
Analytic cond. $29.8309$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 6·19-s − 4·25-s + 8·29-s + 32-s − 6·38-s − 16·41-s + 12·43-s + 2·49-s − 4·50-s + 8·53-s + 8·58-s + 8·59-s + 12·61-s + 64-s + 16·71-s + 16·73-s − 6·76-s − 16·82-s + 12·86-s + 2·98-s − 4·100-s + 8·106-s − 8·107-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/4·16-s − 1.37·19-s − 4/5·25-s + 1.48·29-s + 0.176·32-s − 0.973·38-s − 2.49·41-s + 1.82·43-s + 2/7·49-s − 0.565·50-s + 1.09·53-s + 1.05·58-s + 1.04·59-s + 1.53·61-s + 1/8·64-s + 1.89·71-s + 1.87·73-s − 0.688·76-s − 1.76·82-s + 1.29·86-s + 0.202·98-s − 2/5·100-s + 0.777·106-s − 0.773·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(467856\)    =    \(2^{4} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(29.8309\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 467856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.814410379\)
\(L(\frac12)\) \(\approx\) \(2.814410379\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3 \( 1 \)
19$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \)
79$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
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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

−8.449214706675140374577137305507, −8.163799071418858750785983823329, −7.66178955572243648409098806269, −6.93273577938547297289305636557, −6.67161708699562132524363177791, −6.37452483206806356490987912829, −5.59453942721608634726857267402, −5.32840831471856328566334839582, −4.75211834002268501330498127169, −4.07007049787849789478087703396, −3.87857022401287507201063255664, −3.11735200948967478070120539529, −2.35304085693774311916549110742, −2.00245026895425214068587158329, −0.800785425372416660468654362251, 0.800785425372416660468654362251, 2.00245026895425214068587158329, 2.35304085693774311916549110742, 3.11735200948967478070120539529, 3.87857022401287507201063255664, 4.07007049787849789478087703396, 4.75211834002268501330498127169, 5.32840831471856328566334839582, 5.59453942721608634726857267402, 6.37452483206806356490987912829, 6.67161708699562132524363177791, 6.93273577938547297289305636557, 7.66178955572243648409098806269, 8.163799071418858750785983823329, 8.449214706675140374577137305507

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