Properties

Label 2-3-1.1-c29-0-3
Degree $2$
Conductor $3$
Sign $-1$
Analytic cond. $15.9834$
Root an. cond. $3.99792$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.08e4·2-s − 4.78e6·3-s − 4.19e8·4-s + 2.20e10·5-s + 5.19e10·6-s − 6.70e11·7-s + 1.03e13·8-s + 2.28e13·9-s − 2.38e14·10-s + 1.26e15·11-s + 2.00e15·12-s − 2.39e16·13-s + 7.28e15·14-s − 1.05e17·15-s + 1.12e17·16-s − 6.62e17·17-s − 2.48e17·18-s − 4.89e18·19-s − 9.21e18·20-s + 3.20e18·21-s − 1.37e19·22-s − 1.50e19·23-s − 4.96e19·24-s + 2.97e20·25-s + 2.60e20·26-s − 1.09e20·27-s + 2.81e20·28-s + ⋯
L(s)  = 1  − 0.468·2-s − 0.577·3-s − 0.780·4-s + 1.61·5-s + 0.270·6-s − 0.373·7-s + 0.834·8-s + 0.333·9-s − 0.755·10-s + 1.00·11-s + 0.450·12-s − 1.68·13-s + 0.175·14-s − 0.930·15-s + 0.389·16-s − 0.954·17-s − 0.156·18-s − 1.40·19-s − 1.25·20-s + 0.215·21-s − 0.471·22-s − 0.270·23-s − 0.481·24-s + 1.59·25-s + 0.790·26-s − 0.192·27-s + 0.291·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Analytic conductor: \(15.9834\)
Root analytic conductor: \(3.99792\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :29/2),\ -1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 4.78e6T \)
good2 \( 1 + 1.08e4T + 5.36e8T^{2} \)
5 \( 1 - 2.20e10T + 1.86e20T^{2} \)
7 \( 1 + 6.70e11T + 3.21e24T^{2} \)
11 \( 1 - 1.26e15T + 1.58e30T^{2} \)
13 \( 1 + 2.39e16T + 2.01e32T^{2} \)
17 \( 1 + 6.62e17T + 4.81e35T^{2} \)
19 \( 1 + 4.89e18T + 1.21e37T^{2} \)
23 \( 1 + 1.50e19T + 3.09e39T^{2} \)
29 \( 1 - 3.99e20T + 2.56e42T^{2} \)
31 \( 1 - 1.20e21T + 1.77e43T^{2} \)
37 \( 1 + 9.70e22T + 3.00e45T^{2} \)
41 \( 1 - 6.14e22T + 5.89e46T^{2} \)
43 \( 1 - 1.07e23T + 2.34e47T^{2} \)
47 \( 1 + 1.01e24T + 3.09e48T^{2} \)
53 \( 1 + 1.63e25T + 1.00e50T^{2} \)
59 \( 1 + 1.58e25T + 2.26e51T^{2} \)
61 \( 1 + 3.52e25T + 5.95e51T^{2} \)
67 \( 1 - 1.44e26T + 9.04e52T^{2} \)
71 \( 1 - 4.44e26T + 4.85e53T^{2} \)
73 \( 1 + 3.43e26T + 1.08e54T^{2} \)
79 \( 1 + 5.48e27T + 1.07e55T^{2} \)
83 \( 1 - 5.16e27T + 4.50e55T^{2} \)
89 \( 1 - 3.23e28T + 3.40e56T^{2} \)
97 \( 1 - 5.22e27T + 4.13e57T^{2} \)
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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

−17.54325833699976025160366699611, −17.10835813026695821590247734818, −14.24786298158883391104484561517, −12.76971412813786345004273270846, −10.21597040378714655350991290085, −9.179276186295923644281376754626, −6.49060567627457641714563629186, −4.78248232069624325728104945710, −1.84917962322803316114029015551, 0, 1.84917962322803316114029015551, 4.78248232069624325728104945710, 6.49060567627457641714563629186, 9.179276186295923644281376754626, 10.21597040378714655350991290085, 12.76971412813786345004273270846, 14.24786298158883391104484561517, 17.10835813026695821590247734818, 17.54325833699976025160366699611

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