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Properties

Label 2-3-3.2-c36-0-2
Degree $2$
Conductor $3$
Sign $-0.138 + 0.990i$
Analytic cond. $24.6273$
Root an. cond. $4.96259$
Motivic weight $36$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.02e5i·2-s + (−5.37e7 + 3.83e8i)3-s − 2.28e10·4-s + 4.03e12i·5-s + (−1.16e14 − 1.62e13i)6-s + 4.01e14·7-s + 1.38e16i·8-s + (−1.44e17 − 4.12e16i)9-s − 1.21e18·10-s + 3.25e18i·11-s + (1.22e18 − 8.75e18i)12-s − 1.15e20·13-s + 1.21e20i·14-s + (−1.54e21 − 2.16e20i)15-s − 5.77e21·16-s + 1.35e22i·17-s + ⋯
L(s)  = 1  + 1.15i·2-s + (−0.138 + 0.990i)3-s − 0.332·4-s + 1.05i·5-s + (−1.14 − 0.160i)6-s + 0.246·7-s + 0.770i·8-s + (−0.961 − 0.274i)9-s − 1.21·10-s + 0.585i·11-s + (0.0461 − 0.328i)12-s − 1.02·13-s + 0.284i·14-s + (−1.04 − 0.146i)15-s − 1.22·16-s + 0.965i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & (-0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.138 + 0.990i$
Analytic conductor: \(24.6273\)
Root analytic conductor: \(4.96259\)
Motivic weight: \(36\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :18),\ -0.138 + 0.990i)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(1.512546049\)
\(L(\frac12)\) \(\approx\) \(1.512546049\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.37e7 - 3.83e8i)T \)
good2 \( 1 - 3.02e5iT - 6.87e10T^{2} \)
5 \( 1 - 4.03e12iT - 1.45e25T^{2} \)
7 \( 1 - 4.01e14T + 2.65e30T^{2} \)
11 \( 1 - 3.25e18iT - 3.09e37T^{2} \)
13 \( 1 + 1.15e20T + 1.26e40T^{2} \)
17 \( 1 - 1.35e22iT - 1.97e44T^{2} \)
19 \( 1 - 1.84e23T + 1.08e46T^{2} \)
23 \( 1 + 3.27e24iT - 1.05e49T^{2} \)
29 \( 1 + 2.12e26iT - 4.42e52T^{2} \)
31 \( 1 - 1.84e25T + 4.88e53T^{2} \)
37 \( 1 - 3.11e28T + 2.85e56T^{2} \)
41 \( 1 - 2.06e29iT - 1.14e58T^{2} \)
43 \( 1 + 1.49e29T + 6.38e58T^{2} \)
47 \( 1 + 1.35e29iT - 1.56e60T^{2} \)
53 \( 1 - 1.17e30iT - 1.18e62T^{2} \)
59 \( 1 + 4.51e31iT - 5.63e63T^{2} \)
61 \( 1 + 1.40e32T + 1.87e64T^{2} \)
67 \( 1 - 3.01e32T + 5.47e65T^{2} \)
71 \( 1 - 1.78e33iT - 4.41e66T^{2} \)
73 \( 1 + 4.43e33T + 1.20e67T^{2} \)
79 \( 1 - 1.68e34T + 2.06e68T^{2} \)
83 \( 1 + 3.28e34iT - 1.22e69T^{2} \)
89 \( 1 - 2.03e34iT - 1.50e70T^{2} \)
97 \( 1 + 9.19e35T + 3.34e71T^{2} \)
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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

−17.89394873153397213950548410865, −16.58214814139087710492790508983, −15.10406713732621319205090463113, −14.48108319706892555650423726979, −11.42060735138986702717471454937, −9.872569097821498153211605325271, −7.79277642042513808308354926824, −6.28990950478170886453390461502, −4.75557280413119842863514083162, −2.73301557727581624006663229202, 0.52958644252986728793728414214, 1.43750552745232256077835493214, 2.94698797465139545560106051261, 5.20268147775393257326850863668, 7.41289101260352769869952448552, 9.325630060394530624024058736127, 11.45471832575604412872859689377, 12.37190811185708290988876130459, 13.68152877397768991320297473972, 16.40494856870854492432026002902

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