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Properties

Label 2-3-3.2-c66-0-9
Degree $2$
Conductor $3$
Sign $-0.983 - 0.182i$
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  + 4.54e9i·2-s + (5.46e15 + 1.01e15i)3-s + 5.31e19·4-s + 1.75e23i·5-s + (−4.61e24 + 2.48e25i)6-s − 8.19e27·7-s + 5.76e29i·8-s + (2.88e31 + 1.11e31i)9-s − 7.95e32·10-s + 2.37e34i·11-s + (2.90e35 + 5.40e34i)12-s + 7.13e36·13-s − 3.71e37i·14-s + (−1.78e38 + 9.58e38i)15-s + 1.30e39·16-s + 5.94e40i·17-s + ⋯
L(s)  = 1  + 0.528i·2-s + (0.983 + 0.182i)3-s + 0.720·4-s + 1.50i·5-s + (−0.0966 + 0.519i)6-s − 1.05·7-s + 0.909i·8-s + (0.933 + 0.359i)9-s − 0.795·10-s + 1.02i·11-s + (0.708 + 0.131i)12-s + 1.23·13-s − 0.560i·14-s + (−0.275 + 1.48i)15-s + 0.240·16-s + 1.47i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(67-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+33) \, L(s)\cr =\mathstrut & (-0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.983 - 0.182i$
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: \((2,\ 3,\ (\ :33),\ -0.983 - 0.182i)\)

Particular Values

\(L(\frac{67}{2})\) \(\approx\) \(3.721082326\)
\(L(\frac12)\) \(\approx\) \(3.721082326\)
\(L(34)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.46e15 - 1.01e15i)T \)
good2 \( 1 - 4.54e9iT - 7.37e19T^{2} \)
5 \( 1 - 1.75e23iT - 1.35e46T^{2} \)
7 \( 1 + 8.19e27T + 5.97e55T^{2} \)
11 \( 1 - 2.37e34iT - 5.39e68T^{2} \)
13 \( 1 - 7.13e36T + 3.31e73T^{2} \)
17 \( 1 - 5.94e40iT - 1.62e81T^{2} \)
19 \( 1 - 8.13e41T + 2.49e84T^{2} \)
23 \( 1 + 1.37e45iT - 7.48e89T^{2} \)
29 \( 1 + 9.76e47iT - 3.29e96T^{2} \)
31 \( 1 + 1.85e49T + 2.69e98T^{2} \)
37 \( 1 - 6.12e51T + 3.17e103T^{2} \)
41 \( 1 - 1.20e53iT - 2.77e106T^{2} \)
43 \( 1 + 2.91e53T + 6.44e107T^{2} \)
47 \( 1 + 1.66e54iT - 2.28e110T^{2} \)
53 \( 1 + 6.92e56iT - 6.34e113T^{2} \)
59 \( 1 + 2.60e58iT - 7.52e116T^{2} \)
61 \( 1 - 1.11e59T + 6.78e117T^{2} \)
67 \( 1 - 3.96e59T + 3.31e120T^{2} \)
71 \( 1 + 1.42e61iT - 1.52e122T^{2} \)
73 \( 1 + 2.22e61T + 9.53e122T^{2} \)
79 \( 1 + 3.86e62T + 1.75e125T^{2} \)
83 \( 1 - 9.18e62iT - 4.56e126T^{2} \)
89 \( 1 - 3.59e64iT - 4.56e128T^{2} \)
97 \( 1 + 5.76e65T + 1.33e131T^{2} \)
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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

−14.51974998773165908082091719816, −12.90485065760264586374074451874, −10.89761800429581062318530533043, −9.932561970947997023307737586624, −8.160593769973988229956671533135, −6.91852591160039405545271771914, −6.25588061921126236059699854627, −3.80356020921714812886520557754, −2.85353845441109223133671342196, −1.87755119236548036124540485425, 0.71400622360229106883414514458, 1.41468659267835547926094505554, 2.96463008317429504944007540461, 3.76248354541741762575500234510, 5.73745534206022690176910619036, 7.30578808293626948236214557942, 8.804834016833050100335662516488, 9.636393567183491207904689883284, 11.51278833989287835869317614138, 12.87059619014701935920057929851

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