Properties

Label 2-3-1.1-c23-0-1
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $10.0561$
Root an. cond. $3.17113$
Motivic weight $23$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56e3·2-s − 1.77e5·3-s − 5.94e6·4-s + 1.13e8·5-s − 2.77e8·6-s + 7.85e9·7-s − 2.24e10·8-s + 3.13e10·9-s + 1.77e11·10-s + 1.03e12·11-s + 1.05e12·12-s + 8.08e12·13-s + 1.22e13·14-s − 2.01e13·15-s + 1.48e13·16-s − 1.38e14·17-s + 4.90e13·18-s − 1.41e14·19-s − 6.75e14·20-s − 1.39e15·21-s + 1.61e15·22-s + 4.80e15·23-s + 3.97e15·24-s + 9.90e14·25-s + 1.26e16·26-s − 5.55e15·27-s − 4.66e16·28-s + ⋯
L(s)  = 1  + 0.539·2-s − 0.577·3-s − 0.708·4-s + 1.04·5-s − 0.311·6-s + 1.50·7-s − 0.922·8-s + 0.333·9-s + 0.561·10-s + 1.08·11-s + 0.409·12-s + 1.25·13-s + 0.810·14-s − 0.600·15-s + 0.210·16-s − 0.981·17-s + 0.179·18-s − 0.279·19-s − 0.737·20-s − 0.866·21-s + 0.588·22-s + 1.05·23-s + 0.532·24-s + 0.0831·25-s + 0.675·26-s − 0.192·27-s − 1.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(10.0561\)
Root analytic conductor: \(3.17113\)
Motivic weight: \(23\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(2.208741712\)
\(L(\frac12)\) \(\approx\) \(2.208741712\)
\(L(\frac{25}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.77e5T \)
good2 \( 1 - 1.56e3T + 8.38e6T^{2} \)
5 \( 1 - 1.13e8T + 1.19e16T^{2} \)
7 \( 1 - 7.85e9T + 2.73e19T^{2} \)
11 \( 1 - 1.03e12T + 8.95e23T^{2} \)
13 \( 1 - 8.08e12T + 4.17e25T^{2} \)
17 \( 1 + 1.38e14T + 1.99e28T^{2} \)
19 \( 1 + 1.41e14T + 2.57e29T^{2} \)
23 \( 1 - 4.80e15T + 2.08e31T^{2} \)
29 \( 1 + 1.45e16T + 4.31e33T^{2} \)
31 \( 1 + 8.10e16T + 2.00e34T^{2} \)
37 \( 1 - 1.19e18T + 1.17e36T^{2} \)
41 \( 1 - 6.63e18T + 1.24e37T^{2} \)
43 \( 1 + 1.06e19T + 3.71e37T^{2} \)
47 \( 1 - 1.16e19T + 2.87e38T^{2} \)
53 \( 1 + 7.56e19T + 4.55e39T^{2} \)
59 \( 1 + 4.19e19T + 5.36e40T^{2} \)
61 \( 1 - 7.45e19T + 1.15e41T^{2} \)
67 \( 1 + 1.29e21T + 9.99e41T^{2} \)
71 \( 1 + 2.37e21T + 3.79e42T^{2} \)
73 \( 1 + 3.03e21T + 7.18e42T^{2} \)
79 \( 1 - 3.75e20T + 4.42e43T^{2} \)
83 \( 1 + 9.99e21T + 1.37e44T^{2} \)
89 \( 1 + 5.36e21T + 6.85e44T^{2} \)
97 \( 1 - 1.11e22T + 4.96e45T^{2} \)
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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

−21.02346685635690036292227301379, −18.16776800142464036292479820873, −17.31607371478141726122240935026, −14.62677916772822452245415391676, −13.31147812131123570052965132983, −11.24589917709964709030554567062, −8.966255851812802516898269592285, −5.96245679305929442058441056383, −4.44630644229121449340527660913, −1.38131907346156944497222112369, 1.38131907346156944497222112369, 4.44630644229121449340527660913, 5.96245679305929442058441056383, 8.966255851812802516898269592285, 11.24589917709964709030554567062, 13.31147812131123570052965132983, 14.62677916772822452245415391676, 17.31607371478141726122240935026, 18.16776800142464036292479820873, 21.02346685635690036292227301379

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