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Properties

Label 2-6-3.2-c6-0-0
Degree $2$
Conductor $6$
Sign $0.628 - 0.777i$
Analytic cond. $1.38032$
Root an. cond. $1.17487$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.65i·2-s + (21 + 16.9i)3-s − 32.0·4-s − 169. i·5-s + (−96 + 118. i)6-s + 2·7-s − 181. i·8-s + (153. + 712. i)9-s + 960.·10-s − 33.9i·11-s + (−672. − 543. i)12-s − 2.95e3·13-s + 11.3i·14-s + (2.88e3 − 3.56e3i)15-s + 1.02e3·16-s + 4.48e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.777 + 0.628i)3-s − 0.500·4-s − 1.35i·5-s + (−0.444 + 0.549i)6-s + 0.00583·7-s − 0.353i·8-s + (0.209 + 0.977i)9-s + 0.960·10-s − 0.0255i·11-s + (−0.388 − 0.314i)12-s − 1.34·13-s + 0.00412i·14-s + (0.853 − 1.05i)15-s + 0.250·16-s + 0.911i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(1.38032\)
Root analytic conductor: \(1.17487\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :3),\ 0.628 - 0.777i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.14628 + 0.547455i\)
\(L(\frac12)\) \(\approx\) \(1.14628 + 0.547455i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 5.65iT \)
3 \( 1 + (-21 - 16.9i)T \)
good5 \( 1 + 169. iT - 1.56e4T^{2} \)
7 \( 1 - 2T + 1.17e5T^{2} \)
11 \( 1 + 33.9iT - 1.77e6T^{2} \)
13 \( 1 + 2.95e3T + 4.82e6T^{2} \)
17 \( 1 - 4.48e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.25e3T + 4.70e7T^{2} \)
23 \( 1 + 1.02e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.20e3iT - 5.94e8T^{2} \)
31 \( 1 - 2.28e4T + 8.87e8T^{2} \)
37 \( 1 - 3.40e4T + 2.56e9T^{2} \)
41 \( 1 + 1.67e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.40e3T + 6.32e9T^{2} \)
47 \( 1 - 1.79e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.92e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.26e5iT - 4.21e10T^{2} \)
61 \( 1 + 6.25e4T + 5.15e10T^{2} \)
67 \( 1 - 4.38e5T + 9.04e10T^{2} \)
71 \( 1 + 6.82e4iT - 1.28e11T^{2} \)
73 \( 1 + 7.30e5T + 1.51e11T^{2} \)
79 \( 1 - 3.40e5T + 2.43e11T^{2} \)
83 \( 1 - 4.96e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.86e5iT - 4.96e11T^{2} \)
97 \( 1 + 2.81e5T + 8.32e11T^{2} \)
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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

−22.13373863258863335150093069922, −20.72176959406186343523486861247, −19.46638715118433574189026732750, −17.12696437855432872860112574169, −15.99713022351561678227859458465, −14.49119320717986983194720810727, −12.78562237311126828980232507898, −9.617620391103364735432910615057, −8.159996850570753693766998243183, −4.78440219298001570693256558400, 2.81060547486885503373252387275, 7.32356439514048219557212455775, 9.782251679734233016659644703039, 11.81246219455376496576037678043, 13.73957209498856935585648521689, 14.91387849581282328263842151229, 17.87641256636620115997628881121, 18.93848836651635313512249730379, 20.05673684917198431363307465258, 21.75047343802629783924738298655

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