Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 28.1·5-s − 151.·7-s + 277.·11-s + 583.·13-s + 1.61e3·17-s + 1.36e3·19-s − 856.·23-s − 2.33e3·25-s − 8.53e3·29-s + 2.93e3·31-s + 4.26e3·35-s + 4.03e3·37-s − 1.88e4·41-s − 2.03e4·43-s + 295.·47-s + 6.11e3·49-s − 3.03e3·53-s − 7.81e3·55-s + 1.72e4·59-s + 2.56e4·61-s − 1.64e4·65-s − 2.62e4·67-s − 7.66e4·71-s + 1.49e3·73-s − 4.20e4·77-s + 9.92e4·79-s − 5.00e4·83-s + ⋯
L(s)  = 1  − 0.503·5-s − 1.16·7-s + 0.692·11-s + 0.958·13-s + 1.35·17-s + 0.869·19-s − 0.337·23-s − 0.746·25-s − 1.88·29-s + 0.549·31-s + 0.587·35-s + 0.484·37-s − 1.75·41-s − 1.67·43-s + 0.0195·47-s + 0.363·49-s − 0.148·53-s − 0.348·55-s + 0.644·59-s + 0.882·61-s − 0.482·65-s − 0.715·67-s − 1.80·71-s + 0.0328·73-s − 0.808·77-s + 1.78·79-s − 0.797·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(5\)
character  :  $\chi_{324} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 324,\ (\ :5/2),\ -1)\)
\(L(3)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 28.1T + 3.12e3T^{2} \)
7 \( 1 + 151.T + 1.68e4T^{2} \)
11 \( 1 - 277.T + 1.61e5T^{2} \)
13 \( 1 - 583.T + 3.71e5T^{2} \)
17 \( 1 - 1.61e3T + 1.41e6T^{2} \)
19 \( 1 - 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + 856.T + 6.43e6T^{2} \)
29 \( 1 + 8.53e3T + 2.05e7T^{2} \)
31 \( 1 - 2.93e3T + 2.86e7T^{2} \)
37 \( 1 - 4.03e3T + 6.93e7T^{2} \)
41 \( 1 + 1.88e4T + 1.15e8T^{2} \)
43 \( 1 + 2.03e4T + 1.47e8T^{2} \)
47 \( 1 - 295.T + 2.29e8T^{2} \)
53 \( 1 + 3.03e3T + 4.18e8T^{2} \)
59 \( 1 - 1.72e4T + 7.14e8T^{2} \)
61 \( 1 - 2.56e4T + 8.44e8T^{2} \)
67 \( 1 + 2.62e4T + 1.35e9T^{2} \)
71 \( 1 + 7.66e4T + 1.80e9T^{2} \)
73 \( 1 - 1.49e3T + 2.07e9T^{2} \)
79 \( 1 - 9.92e4T + 3.07e9T^{2} \)
83 \( 1 + 5.00e4T + 3.93e9T^{2} \)
89 \( 1 + 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + 6.66e4T + 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.10694538224223909875111251455, −9.532659765890703252359559387414, −8.410233923912375327821217637923, −7.41081382314139163127909805264, −6.39586654119470725133106153065, −5.48349930575854184872271367271, −3.81389466556305603483761432502, −3.28937023515789750104612842394, −1.40527025805441690791528503336, 0, 1.40527025805441690791528503336, 3.28937023515789750104612842394, 3.81389466556305603483761432502, 5.48349930575854184872271367271, 6.39586654119470725133106153065, 7.41081382314139163127909805264, 8.410233923912375327821217637923, 9.532659765890703252359559387414, 10.10694538224223909875111251455

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