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Properties

Label 2-6-3.2-c4-0-0
Degree $2$
Conductor $6$
Sign $0.942 - 0.333i$
Analytic cond. $0.620219$
Root an. cond. $0.787540$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·2-s + (−3 − 8.48i)3-s − 8.00·4-s + 16.9i·5-s + (24 − 8.48i)6-s + 26·7-s − 22.6i·8-s + (−62.9 + 50.9i)9-s − 48·10-s − 118. i·11-s + (24.0 + 67.8i)12-s + 50·13-s + 73.5i·14-s + (143. − 50.9i)15-s + 64.0·16-s + 203. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.333 − 0.942i)3-s − 0.500·4-s + 0.678i·5-s + (0.666 − 0.235i)6-s + 0.530·7-s − 0.353i·8-s + (−0.777 + 0.628i)9-s − 0.479·10-s − 0.981i·11-s + (0.166 + 0.471i)12-s + 0.295·13-s + 0.375i·14-s + (0.639 − 0.226i)15-s + 0.250·16-s + 0.704i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.942 - 0.333i$
Analytic conductor: \(0.620219\)
Root analytic conductor: \(0.787540\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :2),\ 0.942 - 0.333i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.783507 + 0.134428i\)
\(L(\frac12)\) \(\approx\) \(0.783507 + 0.134428i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
3 \( 1 + (3 + 8.48i)T \)
good5 \( 1 - 16.9iT - 625T^{2} \)
7 \( 1 - 26T + 2.40e3T^{2} \)
11 \( 1 + 118. iT - 1.46e4T^{2} \)
13 \( 1 - 50T + 2.85e4T^{2} \)
17 \( 1 - 203. iT - 8.35e4T^{2} \)
19 \( 1 + 358T + 1.30e5T^{2} \)
23 \( 1 - 373. iT - 2.79e5T^{2} \)
29 \( 1 + 1.44e3iT - 7.07e5T^{2} \)
31 \( 1 + 742T + 9.23e5T^{2} \)
37 \( 1 - 1.87e3T + 1.87e6T^{2} \)
41 \( 1 - 2.40e3iT - 2.82e6T^{2} \)
43 \( 1 + 262T + 3.41e6T^{2} \)
47 \( 1 + 1.69e3iT - 4.87e6T^{2} \)
53 \( 1 + 458. iT - 7.89e6T^{2} \)
59 \( 1 + 1.81e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.48e3T + 1.38e7T^{2} \)
67 \( 1 + 4.48e3T + 2.01e7T^{2} \)
71 \( 1 - 3.56e3iT - 2.54e7T^{2} \)
73 \( 1 - 290T + 2.83e7T^{2} \)
79 \( 1 - 9.81e3T + 3.89e7T^{2} \)
83 \( 1 - 7.11e3iT - 4.74e7T^{2} \)
89 \( 1 + 7.84e3iT - 6.27e7T^{2} \)
97 \( 1 + 478T + 8.85e7T^{2} \)
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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

−23.17024447456488056848581217564, −21.68275597311725115021911028949, −19.25357393362154487622157535866, −18.14684445262715483101845034384, −16.81785818390148454645443886698, −14.75892510562098282313330351567, −13.30418587357277222887227752585, −11.17254729309775689129723266706, −8.094922226128406330354448144987, −6.21402325888917114417235131314, 4.72066524241859762569471483450, 9.032356027666629672811491447864, 10.77938771145848122171120937157, 12.45785439001653219150492667215, 14.71009596683573888842759605438, 16.53740745237521699110690249021, 17.98253016469205534137148310505, 20.18943605014334686089890033214, 20.95712969799655834888736135005, 22.34967934401573238287634357848

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