Properties

Degree $2$
Conductor $3$
Sign $1$
Motivic weight $39$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.25e6·2-s − 1.16e9·3-s + 1.02e12·4-s − 6.13e12·5-s − 1.45e15·6-s + 3.89e16·7-s + 5.89e17·8-s + 1.35e18·9-s − 7.68e18·10-s − 1.99e20·11-s − 1.18e21·12-s + 6.42e21·13-s + 4.87e22·14-s + 7.12e21·15-s + 1.77e23·16-s + 1.81e24·17-s + 1.69e24·18-s + 6.82e24·19-s − 6.25e24·20-s − 4.52e25·21-s − 2.49e26·22-s + 4.57e26·23-s − 6.84e26·24-s − 1.78e27·25-s + 8.04e27·26-s − 1.57e27·27-s + 3.97e28·28-s + ⋯
L(s)  = 1  + 1.68·2-s − 0.577·3-s + 1.85·4-s − 0.143·5-s − 0.975·6-s + 1.29·7-s + 1.44·8-s + 0.333·9-s − 0.242·10-s − 0.981·11-s − 1.07·12-s + 1.21·13-s + 2.18·14-s + 0.0829·15-s + 0.587·16-s + 1.84·17-s + 0.563·18-s + 0.791·19-s − 0.266·20-s − 0.745·21-s − 1.65·22-s + 1.27·23-s − 0.834·24-s − 0.979·25-s + 2.05·26-s − 0.192·27-s + 2.39·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Motivic weight: \(39\)
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :39/2),\ 1)\)

Particular Values

\(L(20)\) \(\approx\) \(5.209309315\)
\(L(\frac12)\) \(\approx\) \(5.209309315\)
\(L(\frac{41}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.16e9T \)
good2 \( 1 - 1.25e6T + 5.49e11T^{2} \)
5 \( 1 + 6.13e12T + 1.81e27T^{2} \)
7 \( 1 - 3.89e16T + 9.09e32T^{2} \)
11 \( 1 + 1.99e20T + 4.11e40T^{2} \)
13 \( 1 - 6.42e21T + 2.77e43T^{2} \)
17 \( 1 - 1.81e24T + 9.71e47T^{2} \)
19 \( 1 - 6.82e24T + 7.43e49T^{2} \)
23 \( 1 - 4.57e26T + 1.28e53T^{2} \)
29 \( 1 - 3.57e28T + 1.08e57T^{2} \)
31 \( 1 + 1.66e29T + 1.45e58T^{2} \)
37 \( 1 - 2.76e30T + 1.44e61T^{2} \)
41 \( 1 + 5.74e30T + 7.91e62T^{2} \)
43 \( 1 + 3.87e31T + 5.07e63T^{2} \)
47 \( 1 + 2.60e32T + 1.62e65T^{2} \)
53 \( 1 + 1.46e32T + 1.76e67T^{2} \)
59 \( 1 + 1.10e33T + 1.15e69T^{2} \)
61 \( 1 + 9.76e34T + 4.24e69T^{2} \)
67 \( 1 + 4.32e35T + 1.64e71T^{2} \)
71 \( 1 + 8.50e35T + 1.58e72T^{2} \)
73 \( 1 + 2.18e36T + 4.67e72T^{2} \)
79 \( 1 - 1.19e37T + 1.01e74T^{2} \)
83 \( 1 - 1.67e37T + 6.98e74T^{2} \)
89 \( 1 - 1.45e38T + 1.06e76T^{2} \)
97 \( 1 + 7.79e38T + 3.04e77T^{2} \)
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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

−16.19118832955298008918628071607, −14.81522468805728535591554272751, −13.45959797543461921094243692178, −11.96525737344024458868288953903, −10.88450905266892113272519849528, −7.67905762148281441319403498354, −5.74503436072033728824460289475, −4.86601351642958178363589531726, −3.29631035790004653621081044578, −1.37779641069758828311165351227, 1.37779641069758828311165351227, 3.29631035790004653621081044578, 4.86601351642958178363589531726, 5.74503436072033728824460289475, 7.67905762148281441319403498354, 10.88450905266892113272519849528, 11.96525737344024458868288953903, 13.45959797543461921094243692178, 14.81522468805728535591554272751, 16.19118832955298008918628071607

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