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Properties

Label 2-3-3.2-c26-0-4
Degree $2$
Conductor $3$
Sign $-0.876 - 0.481i$
Analytic cond. $12.8487$
Root an. cond. $3.58452$
Motivic weight $26$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.82e3i·2-s + (1.39e6 + 7.67e5i)3-s − 2.93e7·4-s + 1.01e9i·5-s + (−7.53e9 + 1.37e10i)6-s + 1.32e11·7-s + 3.70e11i·8-s + (1.36e12 + 2.14e12i)9-s − 9.99e12·10-s − 2.72e13i·11-s + (−4.10e13 − 2.25e13i)12-s − 3.10e14·13-s + 1.30e15i·14-s + (−7.81e14 + 1.42e15i)15-s − 5.61e15·16-s − 6.64e15i·17-s + ⋯
L(s)  = 1  + 1.19i·2-s + (0.876 + 0.481i)3-s − 0.437·4-s + 0.833i·5-s + (−0.577 + 1.05i)6-s + 1.37·7-s + 0.674i·8-s + (0.536 + 0.843i)9-s − 0.999·10-s − 0.789i·11-s + (−0.383 − 0.210i)12-s − 1.02·13-s + 1.64i·14-s + (−0.401 + 0.730i)15-s − 1.24·16-s − 0.671i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(12.8487\)
Root analytic conductor: \(3.58452\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :13),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.683207 + 2.66274i\)
\(L(\frac12)\) \(\approx\) \(0.683207 + 2.66274i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.39e6 - 7.67e5i)T \)
good2 \( 1 - 9.82e3iT - 6.71e7T^{2} \)
5 \( 1 - 1.01e9iT - 1.49e18T^{2} \)
7 \( 1 - 1.32e11T + 9.38e21T^{2} \)
11 \( 1 + 2.72e13iT - 1.19e27T^{2} \)
13 \( 1 + 3.10e14T + 9.17e28T^{2} \)
17 \( 1 + 6.64e15iT - 9.81e31T^{2} \)
19 \( 1 + 5.27e15T + 1.76e33T^{2} \)
23 \( 1 + 7.91e17iT - 2.54e35T^{2} \)
29 \( 1 - 1.83e19iT - 1.05e38T^{2} \)
31 \( 1 + 8.35e18T + 5.96e38T^{2} \)
37 \( 1 - 3.99e20T + 5.93e40T^{2} \)
41 \( 1 + 5.66e20iT - 8.55e41T^{2} \)
43 \( 1 - 1.91e19T + 2.95e42T^{2} \)
47 \( 1 + 5.72e21iT - 2.98e43T^{2} \)
53 \( 1 - 1.57e22iT - 6.77e44T^{2} \)
59 \( 1 + 7.56e22iT - 1.10e46T^{2} \)
61 \( 1 - 2.63e23T + 2.62e46T^{2} \)
67 \( 1 + 4.96e23T + 3.00e47T^{2} \)
71 \( 1 - 2.95e23iT - 1.35e48T^{2} \)
73 \( 1 + 1.62e23T + 2.79e48T^{2} \)
79 \( 1 - 1.76e24T + 2.17e49T^{2} \)
83 \( 1 - 3.14e24iT - 7.87e49T^{2} \)
89 \( 1 + 1.04e25iT - 4.83e50T^{2} \)
97 \( 1 - 4.46e25T + 4.52e51T^{2} \)
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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

−20.38678519249843866161004759263, −18.33665795391403209023859901554, −16.49578234680189367883797802518, −14.76106690057112157629768152490, −14.33197946855786441467072703433, −10.91610969536625613439363378738, −8.516979683577333931530286032355, −7.20748907345086955996939139118, −4.92744373654485258440718737828, −2.50949932445348244066723293446, 1.20765816529450927259376872976, 2.23013142129565864475211520375, 4.36091240571991166041507869168, 7.77119807203027648865640322779, 9.585069821976254251506781901494, 11.75709847646984660385218913946, 13.03199190548850964803336924662, 14.93775752750667483410045880819, 17.65791333228392157522314737320, 19.43904606624823784444123298784

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