Properties

Degree $2$
Conductor $3$
Sign $-1$
Motivic weight $35$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.79e5·2-s + 1.29e8·3-s + 4.37e10·4-s − 3.65e11·5-s − 3.60e13·6-s − 6.36e14·7-s − 2.62e15·8-s + 1.66e16·9-s + 1.02e17·10-s + 1.49e17·11-s + 5.64e18·12-s + 5.52e19·13-s + 1.77e20·14-s − 4.71e19·15-s − 7.70e20·16-s + 1.81e21·17-s − 4.66e21·18-s − 3.32e22·19-s − 1.59e22·20-s − 8.21e22·21-s − 4.17e22·22-s + 1.00e24·23-s − 3.38e23·24-s − 2.77e24·25-s − 1.54e25·26-s + 2.15e24·27-s − 2.78e25·28-s + ⋯
L(s)  = 1  − 1.50·2-s + 0.577·3-s + 1.27·4-s − 0.214·5-s − 0.870·6-s − 1.03·7-s − 0.411·8-s + 0.333·9-s + 0.322·10-s + 0.0890·11-s + 0.734·12-s + 1.77·13-s + 1.55·14-s − 0.123·15-s − 0.652·16-s + 0.531·17-s − 0.502·18-s − 1.39·19-s − 0.272·20-s − 0.596·21-s − 0.134·22-s + 1.48·23-s − 0.237·24-s − 0.954·25-s − 2.67·26-s + 0.192·27-s − 1.31·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Motivic weight: \(35\)
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :35/2),\ -1)\)

Particular Values

\(L(18)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{37}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.29e8T \)
good2 \( 1 + 2.79e5T + 3.43e10T^{2} \)
5 \( 1 + 3.65e11T + 2.91e24T^{2} \)
7 \( 1 + 6.36e14T + 3.78e29T^{2} \)
11 \( 1 - 1.49e17T + 2.81e36T^{2} \)
13 \( 1 - 5.52e19T + 9.72e38T^{2} \)
17 \( 1 - 1.81e21T + 1.16e43T^{2} \)
19 \( 1 + 3.32e22T + 5.70e44T^{2} \)
23 \( 1 - 1.00e24T + 4.57e47T^{2} \)
29 \( 1 + 4.49e25T + 1.52e51T^{2} \)
31 \( 1 + 2.23e25T + 1.57e52T^{2} \)
37 \( 1 + 2.91e27T + 7.71e54T^{2} \)
41 \( 1 + 4.18e27T + 2.80e56T^{2} \)
43 \( 1 + 5.84e28T + 1.48e57T^{2} \)
47 \( 1 + 5.23e28T + 3.33e58T^{2} \)
53 \( 1 + 2.64e30T + 2.23e60T^{2} \)
59 \( 1 - 2.44e30T + 9.54e61T^{2} \)
61 \( 1 + 1.37e31T + 3.06e62T^{2} \)
67 \( 1 + 6.70e31T + 8.17e63T^{2} \)
71 \( 1 - 4.91e32T + 6.22e64T^{2} \)
73 \( 1 - 1.37e32T + 1.64e65T^{2} \)
79 \( 1 - 1.07e33T + 2.61e66T^{2} \)
83 \( 1 + 3.27e33T + 1.47e67T^{2} \)
89 \( 1 - 3.27e33T + 1.69e68T^{2} \)
97 \( 1 + 1.46e34T + 3.44e69T^{2} \)
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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

−16.81292288933172053374390668519, −15.54018395149442515258646346459, −13.21601293059975712348746058460, −10.86179711063681571934403597889, −9.384661906935522444193449251363, −8.274855955209209189422562454424, −6.62310711375376777026040628729, −3.48928380747504301354951356939, −1.56292338907903054527095736241, 0, 1.56292338907903054527095736241, 3.48928380747504301354951356939, 6.62310711375376777026040628729, 8.274855955209209189422562454424, 9.384661906935522444193449251363, 10.86179711063681571934403597889, 13.21601293059975712348746058460, 15.54018395149442515258646346459, 16.81292288933172053374390668519

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