Properties

Label 2-3-1.1-c41-0-2
Degree $2$
Conductor $3$
Sign $-1$
Analytic cond. $31.9415$
Root an. cond. $5.65168$
Motivic weight $41$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.34e6·2-s − 3.48e9·3-s − 3.91e11·4-s + 1.06e14·5-s + 4.68e15·6-s − 3.99e17·7-s + 3.48e18·8-s + 1.21e19·9-s − 1.43e20·10-s + 2.59e21·11-s + 1.36e21·12-s − 3.34e22·13-s + 5.37e23·14-s − 3.72e23·15-s − 3.82e24·16-s + 2.78e25·17-s − 1.63e25·18-s + 2.51e26·19-s − 4.18e25·20-s + 1.39e27·21-s − 3.49e27·22-s − 2.03e27·23-s − 1.21e28·24-s − 3.40e28·25-s + 4.49e28·26-s − 4.23e28·27-s + 1.56e29·28-s + ⋯
L(s)  = 1  − 0.906·2-s − 0.577·3-s − 0.178·4-s + 0.501·5-s + 0.523·6-s − 1.89·7-s + 1.06·8-s + 0.333·9-s − 0.454·10-s + 1.16·11-s + 0.102·12-s − 0.488·13-s + 1.71·14-s − 0.289·15-s − 0.790·16-s + 1.66·17-s − 0.302·18-s + 1.53·19-s − 0.0892·20-s + 1.09·21-s − 1.05·22-s − 0.246·23-s − 0.616·24-s − 0.748·25-s + 0.442·26-s − 0.192·27-s + 0.336·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Analytic conductor: \(31.9415\)
Root analytic conductor: \(5.65168\)
Motivic weight: \(41\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :41/2),\ -1)\)

Particular Values

\(L(21)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{43}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3.48e9T \)
good2 \( 1 + 1.34e6T + 2.19e12T^{2} \)
5 \( 1 - 1.06e14T + 4.54e28T^{2} \)
7 \( 1 + 3.99e17T + 4.45e34T^{2} \)
11 \( 1 - 2.59e21T + 4.97e42T^{2} \)
13 \( 1 + 3.34e22T + 4.69e45T^{2} \)
17 \( 1 - 2.78e25T + 2.80e50T^{2} \)
19 \( 1 - 2.51e26T + 2.68e52T^{2} \)
23 \( 1 + 2.03e27T + 6.77e55T^{2} \)
29 \( 1 + 6.39e29T + 9.08e59T^{2} \)
31 \( 1 + 3.99e29T + 1.39e61T^{2} \)
37 \( 1 + 1.48e32T + 1.97e64T^{2} \)
41 \( 1 - 5.94e31T + 1.33e66T^{2} \)
43 \( 1 + 1.25e33T + 9.38e66T^{2} \)
47 \( 1 - 3.54e33T + 3.59e68T^{2} \)
53 \( 1 - 1.54e34T + 4.95e70T^{2} \)
59 \( 1 + 1.46e35T + 4.02e72T^{2} \)
61 \( 1 + 4.04e36T + 1.57e73T^{2} \)
67 \( 1 + 8.80e36T + 7.39e74T^{2} \)
71 \( 1 - 1.53e38T + 7.97e75T^{2} \)
73 \( 1 + 2.36e38T + 2.49e76T^{2} \)
79 \( 1 + 4.62e38T + 6.34e77T^{2} \)
83 \( 1 - 1.09e39T + 4.81e78T^{2} \)
89 \( 1 + 1.60e40T + 8.41e79T^{2} \)
97 \( 1 - 8.97e38T + 2.86e81T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.27731134020911919704747936826, −13.78454184469069281023438949660, −12.21530391491336415627324254040, −9.950068406572632759658882952468, −9.463742439339232452374340233725, −7.19649503812756783822299610539, −5.70406383875546182452588870599, −3.54048434342297253000106235990, −1.23714202555919116186724395528, 0, 1.23714202555919116186724395528, 3.54048434342297253000106235990, 5.70406383875546182452588870599, 7.19649503812756783822299610539, 9.463742439339232452374340233725, 9.950068406572632759658882952468, 12.21530391491336415627324254040, 13.78454184469069281023438949660, 16.27731134020911919704747936826

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