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Properties

Label 2-6-3.2-c26-0-6
Degree $2$
Conductor $6$
Sign $-0.867 + 0.497i$
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 + (7.93e5 + 1.38e6i)3-s − 3.35e7·4-s − 1.58e9i·5-s + (8.01e9 − 4.59e9i)6-s + 2.87e10·7-s + 1.94e11i·8-s + (−1.28e12 + 2.19e12i)9-s − 9.19e12·10-s − 3.05e12i·11-s + (−2.66e13 − 4.64e13i)12-s + 9.27e13·13-s − 1.66e14i·14-s + (2.19e15 − 1.25e15i)15-s + 1.12e15·16-s − 1.38e16i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.497 + 0.867i)3-s − 0.500·4-s − 1.29i·5-s + (0.613 − 0.351i)6-s + 0.297·7-s + 0.353i·8-s + (−0.504 + 0.863i)9-s − 0.919·10-s − 0.0884i·11-s + (−0.248 − 0.433i)12-s + 0.306·13-s − 0.210i·14-s + (1.12 − 0.646i)15-s + 0.250·16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (-0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.867 + 0.497i$
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: \((2,\ 6,\ (\ :13),\ -0.867 + 0.497i)\)

Particular Values

\(L(\frac{27}{2})\) \(\approx\) \(1.292653035\)
\(L(\frac12)\) \(\approx\) \(1.292653035\)
\(L(14)\) 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 \( 1 + (-7.93e5 - 1.38e6i)T \)
good5 \( 1 + 1.58e9iT - 1.49e18T^{2} \)
7 \( 1 - 2.87e10T + 9.38e21T^{2} \)
11 \( 1 + 3.05e12iT - 1.19e27T^{2} \)
13 \( 1 - 9.27e13T + 9.17e28T^{2} \)
17 \( 1 + 1.38e16iT - 9.81e31T^{2} \)
19 \( 1 + 7.11e16T + 1.76e33T^{2} \)
23 \( 1 + 3.71e17iT - 2.54e35T^{2} \)
29 \( 1 + 1.20e19iT - 1.05e38T^{2} \)
31 \( 1 + 2.98e19T + 5.96e38T^{2} \)
37 \( 1 + 4.00e20T + 5.93e40T^{2} \)
41 \( 1 + 1.14e21iT - 8.55e41T^{2} \)
43 \( 1 - 3.02e21T + 2.95e42T^{2} \)
47 \( 1 - 6.86e21iT - 2.98e43T^{2} \)
53 \( 1 + 2.16e22iT - 6.77e44T^{2} \)
59 \( 1 + 8.99e22iT - 1.10e46T^{2} \)
61 \( 1 - 4.94e21T + 2.62e46T^{2} \)
67 \( 1 - 7.08e23T + 3.00e47T^{2} \)
71 \( 1 - 4.06e23iT - 1.35e48T^{2} \)
73 \( 1 - 1.88e24T + 2.79e48T^{2} \)
79 \( 1 + 2.41e24T + 2.17e49T^{2} \)
83 \( 1 - 1.68e25iT - 7.87e49T^{2} \)
89 \( 1 - 8.57e24iT - 4.83e50T^{2} \)
97 \( 1 - 2.54e25T + 4.52e51T^{2} \)
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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

−15.89008170602911574009914240449, −14.12498736948655019821218926000, −12.66047690718788517340093698623, −10.96460393781783050101852112976, −9.345564731518687440559782591431, −8.379684681699479272309368917212, −5.12943523461531689497741424172, −4.04879988531831466614207952585, −2.18655651932216561706715907988, −0.38574444123283862280937051296, 1.83740488275707405602613440527, 3.56079650884158242099086206872, 6.14485016215077945789309007139, 7.22149988947354823455273038389, 8.602494824332889495089787745510, 10.76180452973708138786120815443, 12.81633187745173983926667308978, 14.33652749439841150947933703890, 15.09660934675314332853793809530, 17.34655550639649942040630396550

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