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Properties

Label 2-6-3.2-c26-0-3
Degree $2$
Conductor $6$
Sign $0.258 - 0.965i$
Analytic cond. $25.6975$
Root an. cond. $5.06927$
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  + 5.79e3i·2-s + (1.54e6 + 4.12e5i)3-s − 3.35e7·4-s − 1.23e9i·5-s + (−2.38e9 + 8.92e9i)6-s − 6.52e10·7-s − 1.94e11i·8-s + (2.20e12 + 1.26e12i)9-s + 7.14e12·10-s + 4.72e13i·11-s + (−5.16e13 − 1.38e13i)12-s + 3.70e14·13-s − 3.78e14i·14-s + (5.08e14 − 1.89e15i)15-s + 1.12e15·16-s + 2.98e13i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.965 + 0.258i)3-s − 0.500·4-s − 1.00i·5-s + (−0.182 + 0.683i)6-s − 0.673·7-s − 0.353i·8-s + (0.866 + 0.499i)9-s + 0.714·10-s + 1.36i·11-s + (−0.482 − 0.129i)12-s + 1.22·13-s − 0.476i·14-s + (0.261 − 0.975i)15-s + 0.250·16-s + 0.00301i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.258 - 0.965i$
Analytic conductor: \(25.6975\)
Root analytic conductor: \(5.06927\)
Motivic weight: \(26\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :13),\ 0.258 - 0.965i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(2.623782010\)
\(L(\frac12)\) \(\approx\) \(2.623782010\)
\(L(14)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.79e3iT \)
3 \( 1 + (-1.54e6 - 4.12e5i)T \)
good5 \( 1 + 1.23e9iT - 1.49e18T^{2} \)
7 \( 1 + 6.52e10T + 9.38e21T^{2} \)
11 \( 1 - 4.72e13iT - 1.19e27T^{2} \)
13 \( 1 - 3.70e14T + 9.17e28T^{2} \)
17 \( 1 - 2.98e13iT - 9.81e31T^{2} \)
19 \( 1 - 7.01e16T + 1.76e33T^{2} \)
23 \( 1 - 4.09e17iT - 2.54e35T^{2} \)
29 \( 1 - 1.48e19iT - 1.05e38T^{2} \)
31 \( 1 - 9.71e18T + 5.96e38T^{2} \)
37 \( 1 + 2.55e19T + 5.93e40T^{2} \)
41 \( 1 + 1.47e21iT - 8.55e41T^{2} \)
43 \( 1 + 1.20e21T + 2.95e42T^{2} \)
47 \( 1 + 2.01e20iT - 2.98e43T^{2} \)
53 \( 1 + 1.74e22iT - 6.77e44T^{2} \)
59 \( 1 - 1.75e23iT - 1.10e46T^{2} \)
61 \( 1 - 8.70e22T + 2.62e46T^{2} \)
67 \( 1 - 4.13e23T + 3.00e47T^{2} \)
71 \( 1 + 1.95e24iT - 1.35e48T^{2} \)
73 \( 1 - 2.56e24T + 2.79e48T^{2} \)
79 \( 1 + 7.16e24T + 2.17e49T^{2} \)
83 \( 1 - 1.27e25iT - 7.87e49T^{2} \)
89 \( 1 + 2.12e25iT - 4.83e50T^{2} \)
97 \( 1 + 6.17e25T + 4.52e51T^{2} \)
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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

−16.32626825111423155698163086642, −15.46837695757731257921447034367, −13.79156179128932303526001466546, −12.65030173358099227712166188327, −9.768071203303715374616809720218, −8.740399182648588625390152840333, −7.21925704353942976241665941149, −5.07765717399630931535945358078, −3.56525154183647456625362633788, −1.33719430065491625225697319358, 0.931360643907596556733814780499, 2.82165637318965270851144526243, 3.53952159733582160286083128312, 6.38925978895501722071082773814, 8.261998391240300666041464626316, 9.801622857670628970970768054295, 11.31988385600073040251574101388, 13.28126625132038593431539320818, 14.18234354240762443899971561517, 15.90358294910188850086456118781

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