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Properties

Label 2-3-3.2-c66-0-6
Degree $2$
Conductor $3$
Sign $-0.736 - 0.675i$
Analytic cond. $82.7604$
Root an. cond. $9.09727$
Motivic weight $66$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.76e9i·2-s + (4.09e15 + 3.75e15i)3-s + 5.11e19·4-s − 1.11e23i·5-s + (−1.78e25 + 1.95e25i)6-s + 1.00e28·7-s + 5.94e29i·8-s + (2.66e30 + 3.07e31i)9-s + 5.31e32·10-s + 2.71e34i·11-s + (2.09e35 + 1.92e35i)12-s − 8.16e36·13-s + 4.77e37i·14-s + (4.19e38 − 4.57e38i)15-s + 9.38e38·16-s − 3.36e40i·17-s + ⋯
L(s)  = 1  + 0.554i·2-s + (0.736 + 0.675i)3-s + 0.692·4-s − 0.958i·5-s + (−0.374 + 0.408i)6-s + 1.29·7-s + 0.938i·8-s + (0.0862 + 0.996i)9-s + 0.531·10-s + 1.16i·11-s + (0.510 + 0.468i)12-s − 1.41·13-s + 0.719i·14-s + (0.648 − 0.706i)15-s + 0.172·16-s − 0.835i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.736 - 0.675i$
Analytic conductor: \(82.7604\)
Root analytic conductor: \(9.09727\)
Motivic weight: \(66\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :33),\ -0.736 - 0.675i)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(3.435210110\)
\(L(\frac12)\) \(\approx\) \(3.435210110\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.09e15 - 3.75e15i)T \)
good2 \( 1 - 4.76e9iT - 7.37e19T^{2} \)
5 \( 1 + 1.11e23iT - 1.35e46T^{2} \)
7 \( 1 - 1.00e28T + 5.97e55T^{2} \)
11 \( 1 - 2.71e34iT - 5.39e68T^{2} \)
13 \( 1 + 8.16e36T + 3.31e73T^{2} \)
17 \( 1 + 3.36e40iT - 1.62e81T^{2} \)
19 \( 1 + 2.59e42T + 2.49e84T^{2} \)
23 \( 1 - 7.82e44iT - 7.48e89T^{2} \)
29 \( 1 - 2.09e48iT - 3.29e96T^{2} \)
31 \( 1 + 9.44e48T + 2.69e98T^{2} \)
37 \( 1 - 5.25e51T + 3.17e103T^{2} \)
41 \( 1 - 2.65e53iT - 2.77e106T^{2} \)
43 \( 1 - 7.04e53T + 6.44e107T^{2} \)
47 \( 1 - 5.54e54iT - 2.28e110T^{2} \)
53 \( 1 - 5.01e56iT - 6.34e113T^{2} \)
59 \( 1 + 2.39e58iT - 7.52e116T^{2} \)
61 \( 1 - 5.18e58T + 6.78e117T^{2} \)
67 \( 1 - 1.99e60T + 3.31e120T^{2} \)
71 \( 1 + 1.53e61iT - 1.52e122T^{2} \)
73 \( 1 + 8.33e60T + 9.53e122T^{2} \)
79 \( 1 - 2.10e62T + 1.75e125T^{2} \)
83 \( 1 - 2.23e63iT - 4.56e126T^{2} \)
89 \( 1 + 2.08e62iT - 4.56e128T^{2} \)
97 \( 1 - 1.02e65T + 1.33e131T^{2} \)
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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

−14.51731919321309968222629950116, −12.52113621171521792685871445470, −11.04354547012486545680721336511, −9.478630033443232278544098251595, −8.172509053211805990335381008060, −7.29107439437943214101323618928, −5.08229920374033864776675817286, −4.58687488381421422839228257199, −2.46285504623076245985277787830, −1.62691977392056023388442773195, 0.61212563668496490303518114178, 2.08256567985446905513728970631, 2.52310943402345800952601742967, 3.95071224110137476033030890303, 6.19527024075224086376459487723, 7.33173206897391267251110176182, 8.440080206873965166104675118689, 10.41752871918042685150139518772, 11.38709559311669810414537437929, 12.67167013656219702127051547730

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