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Properties

Label 2-432-3.2-c8-0-40
Degree $2$
Conductor $432$
Sign $1$
Analytic cond. $175.987$
Root an. cond. $13.2660$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.27e3·7-s − 2.06e4·13-s − 1.57e5·19-s + 3.90e5·25-s + 1.80e6·31-s + 2.96e6·37-s − 3.49e6·43-s + 1.24e7·49-s − 3.07e5·61-s − 3.72e7·67-s + 3.90e7·73-s − 7.48e7·79-s − 8.81e7·91-s − 8.21e7·97-s + 2.13e8·103-s + 2.03e8·109-s + ⋯
L(s)  = 1  + 1.77·7-s − 0.722·13-s − 1.21·19-s + 25-s + 1.95·31-s + 1.58·37-s − 1.02·43-s + 2.16·49-s − 0.0222·61-s − 1.85·67-s + 1.37·73-s − 1.92·79-s − 1.28·91-s − 0.927·97-s + 1.89·103-s + 1.43·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(175.987\)
Root analytic conductor: \(13.2660\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: $\chi_{432} (161, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.907343140\)
\(L(\frac12)\) \(\approx\) \(2.907343140\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
7 \( 1 - 4273 T + p^{8} T^{2} \)
11 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
13 \( 1 + 20641 T + p^{8} T^{2} \)
17 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
19 \( 1 + 157967 T + p^{8} T^{2} \)
23 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
29 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
31 \( 1 - 1809406 T + p^{8} T^{2} \)
37 \( 1 - 2964959 T + p^{8} T^{2} \)
41 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
43 \( 1 + 3492194 T + p^{8} T^{2} \)
47 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
53 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
59 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
61 \( 1 + 307393 T + p^{8} T^{2} \)
67 \( 1 + 37296239 T + p^{8} T^{2} \)
71 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
73 \( 1 - 39067199 T + p^{8} T^{2} \)
79 \( 1 + 74894159 T + p^{8} T^{2} \)
83 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
89 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
97 \( 1 + 82132513 T + p^{8} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.923609110070151230183423111272, −8.652384045192595144232824305803, −8.128768983147263398281250331739, −7.18235217717813767247015653195, −6.05273028815959690411082041220, −4.79890905541035367281352250469, −4.43838238222916467441541410462, −2.75587022591182725654080627735, −1.80137250704375459640127498214, −0.75336586359617861022238389161, 0.75336586359617861022238389161, 1.80137250704375459640127498214, 2.75587022591182725654080627735, 4.43838238222916467441541410462, 4.79890905541035367281352250469, 6.05273028815959690411082041220, 7.18235217717813767247015653195, 8.128768983147263398281250331739, 8.652384045192595144232824305803, 9.923609110070151230183423111272

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