Properties

Degree 2
Conductor $ 3^{2} $
Sign $1$
Motivic weight 37
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.20e5·2-s + 1.33e11·4-s + 2.63e12·5-s + 8.34e15·7-s − 1.97e15·8-s + 1.37e18·10-s + 2.00e19·11-s − 1.05e20·13-s + 4.34e21·14-s − 1.93e22·16-s − 1.77e22·17-s + 4.00e23·19-s + 3.52e23·20-s + 1.04e25·22-s + 1.07e25·23-s − 6.58e25·25-s − 5.48e25·26-s + 1.11e27·28-s − 1.59e27·29-s + 4.20e27·31-s − 9.82e27·32-s − 9.21e27·34-s + 2.19e28·35-s + 1.71e29·37-s + 2.08e29·38-s − 5.19e27·40-s + 3.95e29·41-s + ⋯
L(s)  = 1  + 1.40·2-s + 0.972·4-s + 0.308·5-s + 1.93·7-s − 0.0387·8-s + 0.433·10-s + 1.08·11-s − 0.259·13-s + 2.71·14-s − 1.02·16-s − 0.305·17-s + 0.882·19-s + 0.300·20-s + 1.52·22-s + 0.692·23-s − 0.904·25-s − 0.364·26-s + 1.88·28-s − 1.40·29-s + 1.08·31-s − 1.40·32-s − 0.428·34-s + 0.598·35-s + 1.67·37-s + 1.24·38-s − 0.0119·40-s + 0.576·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9\)    =    \(3^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(37\)
character  :  $\chi_{9} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 9,\ (\ :37/2),\ 1)$
$L(19)$  $\approx$  $6.660964795$
$L(\frac12)$  $\approx$  $6.660964795$
$L(\frac{39}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 3$,\(F_p(T)\) is a polynomial of degree 2. If $p = 3$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 - 5.20e5T + 1.37e11T^{2} \)
5 \( 1 - 2.63e12T + 7.27e25T^{2} \)
7 \( 1 - 8.34e15T + 1.85e31T^{2} \)
11 \( 1 - 2.00e19T + 3.40e38T^{2} \)
13 \( 1 + 1.05e20T + 1.64e41T^{2} \)
17 \( 1 + 1.77e22T + 3.36e45T^{2} \)
19 \( 1 - 4.00e23T + 2.06e47T^{2} \)
23 \( 1 - 1.07e25T + 2.42e50T^{2} \)
29 \( 1 + 1.59e27T + 1.28e54T^{2} \)
31 \( 1 - 4.20e27T + 1.51e55T^{2} \)
37 \( 1 - 1.71e29T + 1.05e58T^{2} \)
41 \( 1 - 3.95e29T + 4.70e59T^{2} \)
43 \( 1 - 3.48e29T + 2.74e60T^{2} \)
47 \( 1 - 2.42e30T + 7.37e61T^{2} \)
53 \( 1 - 1.34e32T + 6.28e63T^{2} \)
59 \( 1 + 1.06e33T + 3.32e65T^{2} \)
61 \( 1 + 7.04e32T + 1.14e66T^{2} \)
67 \( 1 - 7.19e33T + 3.67e67T^{2} \)
71 \( 1 + 2.35e34T + 3.13e68T^{2} \)
73 \( 1 - 2.69e33T + 8.76e68T^{2} \)
79 \( 1 - 2.20e34T + 1.63e70T^{2} \)
83 \( 1 - 2.35e35T + 1.01e71T^{2} \)
89 \( 1 + 1.13e35T + 1.34e72T^{2} \)
97 \( 1 - 8.44e36T + 3.24e73T^{2} \)
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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

−13.53764710125188718931694109338, −11.93167422745714698387015765752, −11.23269755983740507032510390823, −9.154159686290879592084257790294, −7.54720713749742113369378399355, −5.93993614889857618017393582226, −4.87716626848294671609516296283, −3.99770135500324232484230150378, −2.36713612551349998305232084055, −1.20008575568362234212282145998, 1.20008575568362234212282145998, 2.36713612551349998305232084055, 3.99770135500324232484230150378, 4.87716626848294671609516296283, 5.93993614889857618017393582226, 7.54720713749742113369378399355, 9.154159686290879592084257790294, 11.23269755983740507032510390823, 11.93167422745714698387015765752, 13.53764710125188718931694109338

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