Properties

Label 2-3-1.1-c41-0-4
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  − 4.67e5·2-s − 3.48e9·3-s − 1.98e12·4-s − 2.77e14·5-s + 1.62e15·6-s + 3.80e17·7-s + 1.95e18·8-s + 1.21e19·9-s + 1.29e20·10-s + 1.78e20·11-s + 6.90e21·12-s + 2.06e22·13-s − 1.77e23·14-s + 9.66e23·15-s + 3.44e24·16-s − 2.32e25·17-s − 5.68e24·18-s + 2.05e26·19-s + 5.48e26·20-s − 1.32e27·21-s − 8.31e25·22-s − 4.66e27·23-s − 6.80e27·24-s + 3.12e28·25-s − 9.64e27·26-s − 4.23e28·27-s − 7.53e29·28-s + ⋯
L(s)  = 1  − 0.315·2-s − 0.577·3-s − 0.900·4-s − 1.29·5-s + 0.181·6-s + 1.80·7-s + 0.598·8-s + 0.333·9-s + 0.409·10-s + 0.0798·11-s + 0.520·12-s + 0.301·13-s − 0.567·14-s + 0.750·15-s + 0.712·16-s − 1.38·17-s − 0.105·18-s + 1.25·19-s + 1.17·20-s − 1.04·21-s − 0.0251·22-s − 0.566·23-s − 0.345·24-s + 0.688·25-s − 0.0948·26-s − 0.192·27-s − 1.62·28-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 \( 1 + 4.67e5T + 2.19e12T^{2} \)
5 \( 1 + 2.77e14T + 4.54e28T^{2} \)
7 \( 1 - 3.80e17T + 4.45e34T^{2} \)
11 \( 1 - 1.78e20T + 4.97e42T^{2} \)
13 \( 1 - 2.06e22T + 4.69e45T^{2} \)
17 \( 1 + 2.32e25T + 2.80e50T^{2} \)
19 \( 1 - 2.05e26T + 2.68e52T^{2} \)
23 \( 1 + 4.66e27T + 6.77e55T^{2} \)
29 \( 1 - 1.57e30T + 9.08e59T^{2} \)
31 \( 1 + 4.51e30T + 1.39e61T^{2} \)
37 \( 1 + 1.32e32T + 1.97e64T^{2} \)
41 \( 1 + 3.66e32T + 1.33e66T^{2} \)
43 \( 1 + 6.46e32T + 9.38e66T^{2} \)
47 \( 1 - 2.68e34T + 3.59e68T^{2} \)
53 \( 1 + 1.90e35T + 4.95e70T^{2} \)
59 \( 1 + 3.62e36T + 4.02e72T^{2} \)
61 \( 1 - 2.16e36T + 1.57e73T^{2} \)
67 \( 1 - 3.15e36T + 7.39e74T^{2} \)
71 \( 1 + 4.01e37T + 7.97e75T^{2} \)
73 \( 1 - 7.75e37T + 2.49e76T^{2} \)
79 \( 1 + 8.36e38T + 6.34e77T^{2} \)
83 \( 1 + 1.97e39T + 4.81e78T^{2} \)
89 \( 1 - 3.68e39T + 8.41e79T^{2} \)
97 \( 1 - 4.15e40T + 2.86e81T^{2} \)
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

−15.66372150565877819617480047559, −14.03274344861509385420085579411, −11.95486943206766288562081448307, −10.86642578261287351713840637660, −8.638567083630994171787083270292, −7.57016448783397434689341346615, −5.01867390629877425993820040050, −4.09730662598594763577981892957, −1.30839260264355861021295564827, 0, 1.30839260264355861021295564827, 4.09730662598594763577981892957, 5.01867390629877425993820040050, 7.57016448783397434689341346615, 8.638567083630994171787083270292, 10.86642578261287351713840637660, 11.95486943206766288562081448307, 14.03274344861509385420085579411, 15.66372150565877819617480047559

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