Properties

Label 4-165564-1.1-c1e2-0-4
Degree $4$
Conductor $165564$
Sign $-1$
Analytic cond. $10.5565$
Root an. cond. $1.80251$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 7-s − 2·13-s + 16-s − 4·19-s − 6·25-s − 28-s − 16·31-s + 6·43-s − 6·49-s − 2·52-s + 64-s − 3·73-s − 4·76-s − 4·79-s + 2·91-s − 16·97-s − 6·100-s + 4·103-s + 20·109-s − 112-s − 14·121-s − 16·124-s + 127-s + 131-s + 4·133-s + 137-s + ⋯
L(s)  = 1  + 1/2·4-s − 0.377·7-s − 0.554·13-s + 1/4·16-s − 0.917·19-s − 6/5·25-s − 0.188·28-s − 2.87·31-s + 0.914·43-s − 6/7·49-s − 0.277·52-s + 1/8·64-s − 0.351·73-s − 0.458·76-s − 0.450·79-s + 0.209·91-s − 1.62·97-s − 3/5·100-s + 0.394·103-s + 1.91·109-s − 0.0944·112-s − 1.27·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(165564\)    =    \(2^{2} \cdot 3^{4} \cdot 7 \cdot 73\)
Sign: $-1$
Analytic conductor: \(10.5565\)
Root analytic conductor: \(1.80251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{165564} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 165564,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
73$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \)
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−9.008396983836892003462418239910, −8.579270692429305602137962801016, −7.88076985901541131314806919086, −7.44790995274526122919965355597, −7.18004920453287839454157425275, −6.45670620902883939975408832444, −6.08476625425280506349604022454, −5.50729342835208729568171099973, −5.04166083260593555185233584254, −4.14637548913859065119247396577, −3.79378613255892486142834173555, −3.02925396898687861194049938184, −2.25895213588299889227053061618, −1.67449141677278248735177683986, 0, 1.67449141677278248735177683986, 2.25895213588299889227053061618, 3.02925396898687861194049938184, 3.79378613255892486142834173555, 4.14637548913859065119247396577, 5.04166083260593555185233584254, 5.50729342835208729568171099973, 6.08476625425280506349604022454, 6.45670620902883939975408832444, 7.18004920453287839454157425275, 7.44790995274526122919965355597, 7.88076985901541131314806919086, 8.579270692429305602137962801016, 9.008396983836892003462418239910

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