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Properties

Label 2-6-3.2-c26-0-7
Degree $2$
Conductor $6$
Sign $-0.847 + 0.531i$
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 + (−8.47e5 − 1.35e6i)3-s − 3.35e7·4-s − 2.16e9i·5-s + (7.82e9 − 4.90e9i)6-s + 1.32e11·7-s − 1.94e11i·8-s + (−1.10e12 + 2.28e12i)9-s + 1.25e13·10-s − 4.60e13i·11-s + (2.84e13 + 4.53e13i)12-s − 1.11e14·13-s + 7.65e14i·14-s + (−2.92e15 + 1.83e15i)15-s + 1.12e15·16-s + 8.56e13i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.531 − 0.847i)3-s − 0.500·4-s − 1.77i·5-s + (0.598 − 0.375i)6-s + 1.36·7-s − 0.353i·8-s + (−0.435 + 0.900i)9-s + 1.25·10-s − 1.33i·11-s + (0.265 + 0.423i)12-s − 0.367·13-s + 0.964i·14-s + (−1.50 + 0.943i)15-s + 0.250·16-s + 0.00864i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.847 + 0.531i$
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: \((2,\ 6,\ (\ :13),\ -0.847 + 0.531i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(1.201909069\)
\(L(\frac12)\) \(\approx\) \(1.201909069\)
\(L(14)\) 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 \( 1 + (8.47e5 + 1.35e6i)T \)
good5 \( 1 + 2.16e9iT - 1.49e18T^{2} \)
7 \( 1 - 1.32e11T + 9.38e21T^{2} \)
11 \( 1 + 4.60e13iT - 1.19e27T^{2} \)
13 \( 1 + 1.11e14T + 9.17e28T^{2} \)
17 \( 1 - 8.56e13iT - 9.81e31T^{2} \)
19 \( 1 - 3.34e16T + 1.76e33T^{2} \)
23 \( 1 + 4.89e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.94e18iT - 1.05e38T^{2} \)
31 \( 1 + 4.29e19T + 5.96e38T^{2} \)
37 \( 1 - 4.58e19T + 5.93e40T^{2} \)
41 \( 1 - 7.96e20iT - 8.55e41T^{2} \)
43 \( 1 + 1.50e21T + 2.95e42T^{2} \)
47 \( 1 - 2.64e21iT - 2.98e43T^{2} \)
53 \( 1 - 2.66e22iT - 6.77e44T^{2} \)
59 \( 1 + 2.34e22iT - 1.10e46T^{2} \)
61 \( 1 + 8.16e22T + 2.62e46T^{2} \)
67 \( 1 - 9.08e23T + 3.00e47T^{2} \)
71 \( 1 - 5.09e23iT - 1.35e48T^{2} \)
73 \( 1 + 1.03e24T + 2.79e48T^{2} \)
79 \( 1 + 9.95e23T + 2.17e49T^{2} \)
83 \( 1 - 4.25e23iT - 7.87e49T^{2} \)
89 \( 1 + 3.18e25iT - 4.83e50T^{2} \)
97 \( 1 - 6.67e25T + 4.52e51T^{2} \)
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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

−16.32637905875732769171321592444, −14.10519774318470117380763323891, −12.83377416205069223672586774760, −11.46909813015124379271761495055, −8.714019372655064409860923330607, −7.82124916863010121081787138417, −5.67679321088721489871938049568, −4.77189572110779406848904622100, −1.44790994241176929788526488852, −0.45021001871316084996659327434, 1.99157556790693684811760554959, 3.61676421690774760987391098739, 5.16623390261639371964886930587, 7.30224516921476643878474666051, 9.760665282889575404641681203184, 10.87062026601385044177390432822, 11.75809006265436699078621811244, 14.38799831438580137563568798741, 15.13085554257412360416933883960, 17.58254455305459778845415551823

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