Properties

Label 2-3-1.1-c65-0-2
Degree $2$
Conductor $3$
Sign $1$
Analytic cond. $80.2717$
Root an. cond. $8.95944$
Motivic weight $65$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.75e9·2-s + 1.85e15·3-s − 2.93e19·4-s + 9.23e22·5-s + 5.10e24·6-s − 1.32e27·7-s − 1.82e29·8-s + 3.43e30·9-s + 2.54e32·10-s − 1.13e34·11-s − 5.43e34·12-s + 1.26e36·13-s − 3.63e36·14-s + 1.71e38·15-s + 5.78e38·16-s + 1.26e40·17-s + 9.45e39·18-s + 5.39e41·19-s − 2.70e42·20-s − 2.44e42·21-s − 3.13e43·22-s − 2.86e44·23-s − 3.37e44·24-s + 5.81e45·25-s + 3.47e45·26-s + 6.36e45·27-s + 3.87e46·28-s + ⋯
L(s)  = 1  + 0.453·2-s + 0.577·3-s − 0.794·4-s + 1.77·5-s + 0.261·6-s − 0.451·7-s − 0.813·8-s + 0.333·9-s + 0.804·10-s − 1.62·11-s − 0.458·12-s + 0.791·13-s − 0.204·14-s + 1.02·15-s + 0.425·16-s + 1.29·17-s + 0.151·18-s + 1.48·19-s − 1.40·20-s − 0.260·21-s − 0.737·22-s − 1.58·23-s − 0.469·24-s + 2.14·25-s + 0.358·26-s + 0.192·27-s + 0.359·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $1$
Analytic conductor: \(80.2717\)
Root analytic conductor: \(8.95944\)
Motivic weight: \(65\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :65/2),\ 1)\)

Particular Values

\(L(33)\) \(\approx\) \(3.693289467\)
\(L(\frac12)\) \(\approx\) \(3.693289467\)
\(L(\frac{67}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.85e15T \)
good2 \( 1 - 2.75e9T + 3.68e19T^{2} \)
5 \( 1 - 9.23e22T + 2.71e45T^{2} \)
7 \( 1 + 1.32e27T + 8.53e54T^{2} \)
11 \( 1 + 1.13e34T + 4.90e67T^{2} \)
13 \( 1 - 1.26e36T + 2.54e72T^{2} \)
17 \( 1 - 1.26e40T + 9.53e79T^{2} \)
19 \( 1 - 5.39e41T + 1.31e83T^{2} \)
23 \( 1 + 2.86e44T + 3.25e88T^{2} \)
29 \( 1 + 8.89e46T + 1.13e95T^{2} \)
31 \( 1 - 7.20e47T + 8.67e96T^{2} \)
37 \( 1 - 3.10e50T + 8.57e101T^{2} \)
41 \( 1 - 1.45e52T + 6.77e104T^{2} \)
43 \( 1 - 1.60e53T + 1.49e106T^{2} \)
47 \( 1 + 1.74e54T + 4.85e108T^{2} \)
53 \( 1 - 1.10e56T + 1.19e112T^{2} \)
59 \( 1 - 8.05e56T + 1.27e115T^{2} \)
61 \( 1 - 7.28e57T + 1.11e116T^{2} \)
67 \( 1 - 2.30e59T + 4.95e118T^{2} \)
71 \( 1 - 1.89e60T + 2.14e120T^{2} \)
73 \( 1 - 1.33e59T + 1.30e121T^{2} \)
79 \( 1 - 6.07e61T + 2.21e123T^{2} \)
83 \( 1 + 1.13e62T + 5.49e124T^{2} \)
89 \( 1 + 1.42e62T + 5.13e126T^{2} \)
97 \( 1 - 5.82e64T + 1.38e129T^{2} \)
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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

−13.58324443299515816235830741118, −12.72655821173361671628650490438, −10.11370732355369430594917578627, −9.519106278534591008355282383364, −8.010278669307690800621464766552, −5.92979394939576211699525347524, −5.25129037306274752168150383051, −3.43890774386783786363461435079, −2.39107167226987698435720894102, −0.917066031967248182833924348796, 0.917066031967248182833924348796, 2.39107167226987698435720894102, 3.43890774386783786363461435079, 5.25129037306274752168150383051, 5.92979394939576211699525347524, 8.010278669307690800621464766552, 9.519106278534591008355282383364, 10.11370732355369430594917578627, 12.72655821173361671628650490438, 13.58324443299515816235830741118

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