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Properties

Label 2-3-3.2-c26-0-7
Degree $2$
Conductor $3$
Sign $-0.387 + 0.921i$
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  + 6.90e3i·2-s + (6.18e5 − 1.46e6i)3-s + 1.94e7·4-s − 1.04e9i·5-s + (1.01e10 + 4.27e9i)6-s − 1.82e11·7-s + 5.97e11i·8-s + (−1.77e12 − 1.81e12i)9-s + 7.19e12·10-s − 2.00e13i·11-s + (1.20e13 − 2.85e13i)12-s − 1.77e14·13-s − 1.26e15i·14-s + (−1.53e15 − 6.44e14i)15-s − 2.82e15·16-s − 1.19e16i·17-s + ⋯
L(s)  = 1  + 0.843i·2-s + (0.387 − 0.921i)3-s + 0.289·4-s − 0.854i·5-s + (0.777 + 0.326i)6-s − 1.88·7-s + 1.08i·8-s + (−0.699 − 0.714i)9-s + 0.719·10-s − 0.580i·11-s + (0.112 − 0.266i)12-s − 0.587·13-s − 1.59i·14-s + (−0.787 − 0.331i)15-s − 0.627·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.387 + 0.921i$
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: \((2,\ 3,\ (\ :13),\ -0.387 + 0.921i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(0.616438 - 0.928150i\)
\(L(\frac12)\) \(\approx\) \(0.616438 - 0.928150i\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-6.18e5 + 1.46e6i)T \)
good2 \( 1 - 6.90e3iT - 6.71e7T^{2} \)
5 \( 1 + 1.04e9iT - 1.49e18T^{2} \)
7 \( 1 + 1.82e11T + 9.38e21T^{2} \)
11 \( 1 + 2.00e13iT - 1.19e27T^{2} \)
13 \( 1 + 1.77e14T + 9.17e28T^{2} \)
17 \( 1 + 1.19e16iT - 9.81e31T^{2} \)
19 \( 1 + 3.24e15T + 1.76e33T^{2} \)
23 \( 1 - 1.64e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.21e19iT - 1.05e38T^{2} \)
31 \( 1 - 2.60e18T + 5.96e38T^{2} \)
37 \( 1 + 2.51e19T + 5.93e40T^{2} \)
41 \( 1 + 1.30e21iT - 8.55e41T^{2} \)
43 \( 1 - 1.35e21T + 2.95e42T^{2} \)
47 \( 1 - 1.28e21iT - 2.98e43T^{2} \)
53 \( 1 - 4.21e22iT - 6.77e44T^{2} \)
59 \( 1 - 3.86e22iT - 1.10e46T^{2} \)
61 \( 1 + 1.13e23T + 2.62e46T^{2} \)
67 \( 1 - 4.68e23T + 3.00e47T^{2} \)
71 \( 1 + 5.64e23iT - 1.35e48T^{2} \)
73 \( 1 + 1.49e24T + 2.79e48T^{2} \)
79 \( 1 - 4.28e24T + 2.17e49T^{2} \)
83 \( 1 + 1.38e25iT - 7.87e49T^{2} \)
89 \( 1 + 2.05e25iT - 4.83e50T^{2} \)
97 \( 1 + 3.14e25T + 4.52e51T^{2} \)
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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

−19.15499892385424194438084705341, −16.95871948733667821637859273748, −15.78980477386697362858747713221, −13.66770370530760143359275452496, −12.23225612660747927540687443785, −9.098820991903130948547850205164, −7.28082113656834763954391602758, −5.94751226319760263232550670495, −2.76288325790387578438678184213, −0.41607762356751683053723288440, 2.60106259342594876125321299040, 3.61140335828503747284498780471, 6.64940064794950797549015217206, 9.691780001897148171820062177052, 10.60723831996697395651364205702, 12.70804494682875381188111923618, 15.00329973530100520519695820448, 16.35913580793836199748963571666, 19.14927244370629026238445212856, 20.00794264055239271903813499386

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