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  + 2.18e5·2-s + 1.29e8·3-s + 1.34e10·4-s − 9.68e11·5-s + 2.82e13·6-s − 5.65e14·7-s − 4.57e15·8-s + 1.66e16·9-s − 2.11e17·10-s − 1.62e18·11-s + 1.73e18·12-s − 2.52e19·13-s − 1.23e20·14-s − 1.25e20·15-s − 1.46e21·16-s + 4.37e21·17-s + 3.64e21·18-s + 1.72e22·19-s − 1.29e22·20-s − 7.30e22·21-s − 3.54e23·22-s − 5.13e23·23-s − 5.91e23·24-s − 1.97e24·25-s − 5.51e24·26-s + 2.15e24·27-s − 7.57e24·28-s + ⋯
L(s)  = 1  + 1.17·2-s + 0.577·3-s + 0.390·4-s − 0.567·5-s + 0.680·6-s − 0.918·7-s − 0.719·8-s + 0.333·9-s − 0.669·10-s − 0.968·11-s + 0.225·12-s − 0.809·13-s − 1.08·14-s − 0.327·15-s − 1.23·16-s + 1.28·17-s + 0.393·18-s + 0.720·19-s − 0.221·20-s − 0.530·21-s − 1.14·22-s − 0.758·23-s − 0.415·24-s − 0.677·25-s − 0.954·26-s + 0.192·27-s − 0.358·28-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 \( 1 - 2.18e5T + 3.43e10T^{2} \)
5 \( 1 + 9.68e11T + 2.91e24T^{2} \)
7 \( 1 + 5.65e14T + 3.78e29T^{2} \)
11 \( 1 + 1.62e18T + 2.81e36T^{2} \)
13 \( 1 + 2.52e19T + 9.72e38T^{2} \)
17 \( 1 - 4.37e21T + 1.16e43T^{2} \)
19 \( 1 - 1.72e22T + 5.70e44T^{2} \)
23 \( 1 + 5.13e23T + 4.57e47T^{2} \)
29 \( 1 + 3.39e25T + 1.52e51T^{2} \)
31 \( 1 + 9.89e25T + 1.57e52T^{2} \)
37 \( 1 - 4.58e27T + 7.71e54T^{2} \)
41 \( 1 + 2.87e28T + 2.80e56T^{2} \)
43 \( 1 - 6.85e28T + 1.48e57T^{2} \)
47 \( 1 + 4.79e26T + 3.33e58T^{2} \)
53 \( 1 - 2.32e30T + 2.23e60T^{2} \)
59 \( 1 - 5.75e30T + 9.54e61T^{2} \)
61 \( 1 + 2.67e31T + 3.06e62T^{2} \)
67 \( 1 - 7.66e31T + 8.17e63T^{2} \)
71 \( 1 + 5.08e31T + 6.22e64T^{2} \)
73 \( 1 + 2.74e32T + 1.64e65T^{2} \)
79 \( 1 + 2.13e33T + 2.61e66T^{2} \)
83 \( 1 + 2.22e33T + 1.47e67T^{2} \)
89 \( 1 - 2.07e34T + 1.69e68T^{2} \)
97 \( 1 - 1.55e34T + 3.44e69T^{2} \)
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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

−16.04495064671967522951730817371, −14.74964043303869700742126825607, −13.31385302566793651356660744058, −12.13806351546674382447127171779, −9.694634232283678999756323381430, −7.60188097945322164421779557895, −5.55575620762707654186801387240, −3.85875687224132159374955368759, −2.75682651962078438686755235773, 0, 2.75682651962078438686755235773, 3.85875687224132159374955368759, 5.55575620762707654186801387240, 7.60188097945322164421779557895, 9.694634232283678999756323381430, 12.13806351546674382447127171779, 13.31385302566793651356660744058, 14.74964043303869700742126825607, 16.04495064671967522951730817371

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