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Properties

Label 2-3-3.2-c42-0-5
Degree $2$
Conductor $3$
Sign $0.333 - 0.942i$
Analytic cond. $33.5183$
Root an. cond. $5.78950$
Motivic weight $42$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.13e6i·2-s + (−3.48e9 + 9.86e9i)3-s − 1.66e11·4-s − 7.27e14i·5-s + (−2.10e16 − 7.45e15i)6-s − 7.17e17·7-s + 9.04e18i·8-s + (−8.50e19 − 6.87e19i)9-s + 1.55e21·10-s − 5.19e21i·11-s + (5.80e20 − 1.64e21i)12-s + 2.30e23·13-s − 1.53e24i·14-s + (7.17e24 + 2.53e24i)15-s − 2.00e25·16-s + 3.57e25i·17-s + ⋯
L(s)  = 1  + 1.01i·2-s + (−0.333 + 0.942i)3-s − 0.0378·4-s − 1.52i·5-s + (−0.960 − 0.339i)6-s − 1.28·7-s + 0.980i·8-s + (−0.777 − 0.628i)9-s + 1.55·10-s − 0.701i·11-s + (0.0126 − 0.0356i)12-s + 0.931·13-s − 1.30i·14-s + (1.43 + 0.508i)15-s − 1.03·16-s + 0.517i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.333 - 0.942i$
Analytic conductor: \(33.5183\)
Root analytic conductor: \(5.78950\)
Motivic weight: \(42\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :21),\ 0.333 - 0.942i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(1.657594474\)
\(L(\frac12)\) \(\approx\) \(1.657594474\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.48e9 - 9.86e9i)T \)
good2 \( 1 - 2.13e6iT - 4.39e12T^{2} \)
5 \( 1 + 7.27e14iT - 2.27e29T^{2} \)
7 \( 1 + 7.17e17T + 3.11e35T^{2} \)
11 \( 1 + 5.19e21iT - 5.47e43T^{2} \)
13 \( 1 - 2.30e23T + 6.10e46T^{2} \)
17 \( 1 - 3.57e25iT - 4.77e51T^{2} \)
19 \( 1 - 1.20e27T + 5.10e53T^{2} \)
23 \( 1 - 4.53e28iT - 1.55e57T^{2} \)
29 \( 1 + 3.09e30iT - 2.63e61T^{2} \)
31 \( 1 - 2.88e31T + 4.33e62T^{2} \)
37 \( 1 + 7.76e29T + 7.31e65T^{2} \)
41 \( 1 - 1.18e33iT - 5.45e67T^{2} \)
43 \( 1 - 1.46e33T + 4.03e68T^{2} \)
47 \( 1 + 4.50e34iT - 1.69e70T^{2} \)
53 \( 1 + 2.22e36iT - 2.62e72T^{2} \)
59 \( 1 - 3.90e36iT - 2.37e74T^{2} \)
61 \( 1 - 2.78e37T + 9.63e74T^{2} \)
67 \( 1 - 3.05e38T + 4.95e76T^{2} \)
71 \( 1 + 3.64e38iT - 5.66e77T^{2} \)
73 \( 1 - 1.42e39T + 1.81e78T^{2} \)
79 \( 1 - 3.60e38T + 5.01e79T^{2} \)
83 \( 1 + 7.96e39iT - 3.99e80T^{2} \)
89 \( 1 + 6.40e40iT - 7.48e81T^{2} \)
97 \( 1 - 5.27e41T + 2.78e83T^{2} \)
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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

−16.10801875091766915576246407085, −15.80924944524887802300426424027, −13.52563558109262358324083498001, −11.66605714338208955505236056047, −9.584032090709616642664044894812, −8.372468175561931707494604161803, −6.16522051574486351720231153244, −5.23832066071894484927133740510, −3.50685068140775349797812985051, −0.75517479738336487920874185080, 0.856605047580397230726380938777, 2.49960711277215865982690254624, 3.26760080390058574678121160289, 6.36617728561207760595871763966, 7.12419506560109416011582736747, 9.938582683824295958594837622159, 11.14468723496899272152217619587, 12.38771310995479210120179746477, 13.79748237031635691693105750991, 15.90270843279288904424797914028

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