Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s − 7·13-s + 19-s − 5·25-s + 4·31-s − 37-s − 8·43-s + 18·49-s − 13·61-s − 11·67-s + 17·73-s + 13·79-s + 35·91-s + 5·97-s + 7·103-s + 2·109-s + ⋯
L(s)  = 1  − 1.88·7-s − 1.94·13-s + 0.229·19-s − 25-s + 0.718·31-s − 0.164·37-s − 1.21·43-s + 18/7·49-s − 1.66·61-s − 1.34·67-s + 1.98·73-s + 1.46·79-s + 3.66·91-s + 0.507·97-s + 0.689·103-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-1$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -1)\)

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$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + p T^{2} \) 1.5.a
7 \( 1 + 5 T + p T^{2} \) 1.7.f
11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 + 7 T + p T^{2} \) 1.13.h
17 \( 1 + p T^{2} \) 1.17.a
19 \( 1 - T + p T^{2} \) 1.19.ab
23 \( 1 + p T^{2} \) 1.23.a
29 \( 1 + p T^{2} \) 1.29.a
31 \( 1 - 4 T + p T^{2} \) 1.31.ae
37 \( 1 + T + p T^{2} \) 1.37.b
41 \( 1 + p T^{2} \) 1.41.a
43 \( 1 + 8 T + p T^{2} \) 1.43.i
47 \( 1 + p T^{2} \) 1.47.a
53 \( 1 + p T^{2} \) 1.53.a
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 + 13 T + p T^{2} \) 1.61.n
67 \( 1 + 11 T + p T^{2} \) 1.67.l
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 - 17 T + p T^{2} \) 1.73.ar
79 \( 1 - 13 T + p T^{2} \) 1.79.an
83 \( 1 + p T^{2} \) 1.83.a
89 \( 1 + p T^{2} \) 1.89.a
97 \( 1 - 5 T + p T^{2} \) 1.97.af
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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

−10.40150154645553667557841727686, −9.772574318258794189724223366751, −9.220031324604069206185525205678, −7.78745335695197737171573573692, −6.91823062220437394787576967524, −6.08584189960381722201120484886, −4.87832652992641646209905365152, −3.51757494847066237517697434263, −2.49929826070171115553006367440, 0, 2.49929826070171115553006367440, 3.51757494847066237517697434263, 4.87832652992641646209905365152, 6.08584189960381722201120484886, 6.91823062220437394787576967524, 7.78745335695197737171573573692, 9.220031324604069206185525205678, 9.772574318258794189724223366751, 10.40150154645553667557841727686

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