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Properties

Label 2-6-3.2-c18-0-1
Degree $2$
Conductor $6$
Sign $0.983 - 0.183i$
Analytic cond. $12.3231$
Root an. cond. $3.51043$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 362. i·2-s + (−3.61e3 − 1.93e4i)3-s − 1.31e5·4-s + 3.69e6i·5-s + (−7.00e6 + 1.30e6i)6-s + 3.18e7·7-s + 4.74e7i·8-s + (−3.61e8 + 1.39e8i)9-s + 1.33e9·10-s − 7.02e8i·11-s + (4.73e8 + 2.53e9i)12-s + 8.10e9·13-s − 1.15e10i·14-s + (7.14e10 − 1.33e10i)15-s + 1.71e10·16-s + 1.82e11i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.183 − 0.983i)3-s − 0.500·4-s + 1.89i·5-s + (−0.695 + 0.129i)6-s + 0.788·7-s + 0.353i·8-s + (−0.932 + 0.360i)9-s + 1.33·10-s − 0.297i·11-s + (0.0917 + 0.491i)12-s + 0.764·13-s − 0.557i·14-s + (1.85 − 0.347i)15-s + 0.250·16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.983 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.983 - 0.183i$
Analytic conductor: \(12.3231\)
Root analytic conductor: \(3.51043\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :9),\ 0.983 - 0.183i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.49968 + 0.138781i\)
\(L(\frac12)\) \(\approx\) \(1.49968 + 0.138781i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 362. iT \)
3 \( 1 + (3.61e3 + 1.93e4i)T \)
good5 \( 1 - 3.69e6iT - 3.81e12T^{2} \)
7 \( 1 - 3.18e7T + 1.62e15T^{2} \)
11 \( 1 + 7.02e8iT - 5.55e18T^{2} \)
13 \( 1 - 8.10e9T + 1.12e20T^{2} \)
17 \( 1 - 1.82e11iT - 1.40e22T^{2} \)
19 \( 1 - 2.50e11T + 1.04e23T^{2} \)
23 \( 1 - 9.32e11iT - 3.24e24T^{2} \)
29 \( 1 - 8.17e12iT - 2.10e26T^{2} \)
31 \( 1 - 2.87e13T + 6.99e26T^{2} \)
37 \( 1 - 3.99e13T + 1.68e28T^{2} \)
41 \( 1 - 3.68e14iT - 1.07e29T^{2} \)
43 \( 1 + 2.11e14T + 2.52e29T^{2} \)
47 \( 1 - 2.53e14iT - 1.25e30T^{2} \)
53 \( 1 - 2.48e14iT - 1.08e31T^{2} \)
59 \( 1 + 1.21e14iT - 7.50e31T^{2} \)
61 \( 1 - 7.62e14T + 1.36e32T^{2} \)
67 \( 1 + 4.78e16T + 7.40e32T^{2} \)
71 \( 1 + 6.41e16iT - 2.10e33T^{2} \)
73 \( 1 - 7.57e16T + 3.46e33T^{2} \)
79 \( 1 - 1.35e16T + 1.43e34T^{2} \)
83 \( 1 + 2.11e17iT - 3.49e34T^{2} \)
89 \( 1 + 1.30e17iT - 1.22e35T^{2} \)
97 \( 1 - 9.26e17T + 5.77e35T^{2} \)
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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

−18.48111562740521328819986725744, −17.67826506652193396398643415952, −14.75024473366228967394132016278, −13.60226819982695682237118788294, −11.59177586466981742220240934126, −10.66843542186751548939622679113, −7.929462688212163478526248611833, −6.21812247830934507349338685393, −3.17749461110040617873476189407, −1.60603000496686569773600133234, 0.73820954359297864085451684902, 4.44192665180808016114122343240, 5.35665333584062995755861993809, 8.308936127550290206990373150178, 9.465325063171447525821599117693, 11.83813762989031446986257986990, 13.77304833072145531357716590767, 15.65534879781288595506477099742, 16.52354247257674077270947180205, 17.71697900027388773652405488996

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