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Properties

Label 2-3-3.2-c66-0-3
Degree $2$
Conductor $3$
Sign $0.547 - 0.836i$
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  − 9.23e9i·2-s + (−3.04e15 + 4.65e15i)3-s − 1.15e19·4-s − 1.92e23i·5-s + (4.29e25 + 2.81e25i)6-s + 2.12e26·7-s − 5.75e29i·8-s + (−1.23e31 − 2.83e31i)9-s − 1.77e33·10-s + 2.81e34i·11-s + (3.51e34 − 5.36e34i)12-s + 1.93e36·13-s − 1.96e36i·14-s + (8.93e38 + 5.85e38i)15-s − 6.16e39·16-s + 3.10e40i·17-s + ⋯
L(s)  = 1  − 1.07i·2-s + (−0.547 + 0.836i)3-s − 0.156·4-s − 1.65i·5-s + (0.899 + 0.589i)6-s + 0.0275·7-s − 0.907i·8-s + (−0.399 − 0.916i)9-s − 1.77·10-s + 1.21i·11-s + (0.0856 − 0.130i)12-s + 0.336·13-s − 0.0295i·14-s + (1.38 + 0.904i)15-s − 1.13·16-s + 0.770i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.547 - 0.836i$
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: \((2,\ 3,\ (\ :33),\ 0.547 - 0.836i)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(0.2190184284\)
\(L(\frac12)\) \(\approx\) \(0.2190184284\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.04e15 - 4.65e15i)T \)
good2 \( 1 + 9.23e9iT - 7.37e19T^{2} \)
5 \( 1 + 1.92e23iT - 1.35e46T^{2} \)
7 \( 1 - 2.12e26T + 5.97e55T^{2} \)
11 \( 1 - 2.81e34iT - 5.39e68T^{2} \)
13 \( 1 - 1.93e36T + 3.31e73T^{2} \)
17 \( 1 - 3.10e40iT - 1.62e81T^{2} \)
19 \( 1 + 2.58e42T + 2.49e84T^{2} \)
23 \( 1 + 1.28e45iT - 7.48e89T^{2} \)
29 \( 1 + 1.79e48iT - 3.29e96T^{2} \)
31 \( 1 - 2.51e48T + 2.69e98T^{2} \)
37 \( 1 + 3.69e51T + 3.17e103T^{2} \)
41 \( 1 + 3.71e52iT - 2.77e106T^{2} \)
43 \( 1 - 3.83e53T + 6.44e107T^{2} \)
47 \( 1 - 2.78e55iT - 2.28e110T^{2} \)
53 \( 1 + 2.47e56iT - 6.34e113T^{2} \)
59 \( 1 - 4.29e58iT - 7.52e116T^{2} \)
61 \( 1 - 6.76e58T + 6.78e117T^{2} \)
67 \( 1 + 1.21e60T + 3.31e120T^{2} \)
71 \( 1 + 9.25e60iT - 1.52e122T^{2} \)
73 \( 1 - 3.92e61T + 9.53e122T^{2} \)
79 \( 1 + 1.61e62T + 1.75e125T^{2} \)
83 \( 1 - 2.73e63iT - 4.56e126T^{2} \)
89 \( 1 - 9.54e63iT - 4.56e128T^{2} \)
97 \( 1 + 5.23e65T + 1.33e131T^{2} \)
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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

−12.75890289502303325396614423899, −12.26701033249483669160492363030, −10.79807610232863776883060334467, −9.735981310914867871906418184847, −8.581226931132030446963953851574, −6.26231670905948126119292577656, −4.60457224506811646332464496075, −4.09769045238178543160408282245, −2.17743228739546269443144610542, −1.01955182616602796188736573823, 0.05743134604995378383319136210, 1.95452354784448988565101352535, 3.17187988763010224866781723319, 5.50769958303425097069598634466, 6.45850923128315993824467324276, 7.12210531780923587182188163744, 8.316897054458287759351686082471, 10.79020993836032507577390066356, 11.46951817852313074404622848784, 13.59353381802128027447555359444

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