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Properties

Label 2-6-3.2-c18-0-5
Degree $2$
Conductor $6$
Sign $-0.135 + 0.990i$
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 + (1.95e4 + 2.66e3i)3-s − 1.31e5·4-s − 1.30e6i·5-s + (9.63e5 − 7.06e6i)6-s + 3.15e7·7-s + 4.74e7i·8-s + (3.73e8 + 1.03e8i)9-s − 4.72e8·10-s − 3.81e9i·11-s + (−2.55e9 − 3.48e8i)12-s − 5.50e9·13-s − 1.14e10i·14-s + (3.47e9 − 2.54e10i)15-s + 1.71e10·16-s − 1.95e11i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.990 + 0.135i)3-s − 0.500·4-s − 0.668i·5-s + (0.0955 − 0.700i)6-s + 0.782·7-s + 0.353i·8-s + (0.963 + 0.267i)9-s − 0.472·10-s − 1.61i·11-s + (−0.495 − 0.0675i)12-s − 0.519·13-s − 0.553i·14-s + (0.0903 − 0.662i)15-s + 0.250·16-s − 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.135 + 0.990i$
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: \((2,\ 6,\ (\ :9),\ -0.135 + 0.990i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(1.60306 - 1.83661i\)
\(L(\frac12)\) \(\approx\) \(1.60306 - 1.83661i\)
\(L(10)\) 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 \( 1 + (-1.95e4 - 2.66e3i)T \)
good5 \( 1 + 1.30e6iT - 3.81e12T^{2} \)
7 \( 1 - 3.15e7T + 1.62e15T^{2} \)
11 \( 1 + 3.81e9iT - 5.55e18T^{2} \)
13 \( 1 + 5.50e9T + 1.12e20T^{2} \)
17 \( 1 + 1.95e11iT - 1.40e22T^{2} \)
19 \( 1 + 1.56e11T + 1.04e23T^{2} \)
23 \( 1 - 1.49e12iT - 3.24e24T^{2} \)
29 \( 1 - 7.21e12iT - 2.10e26T^{2} \)
31 \( 1 - 3.81e12T + 6.99e26T^{2} \)
37 \( 1 - 2.50e14T + 1.68e28T^{2} \)
41 \( 1 - 4.39e14iT - 1.07e29T^{2} \)
43 \( 1 + 5.41e14T + 2.52e29T^{2} \)
47 \( 1 - 1.94e15iT - 1.25e30T^{2} \)
53 \( 1 + 1.53e15iT - 1.08e31T^{2} \)
59 \( 1 - 6.50e15iT - 7.50e31T^{2} \)
61 \( 1 - 3.87e15T + 1.36e32T^{2} \)
67 \( 1 - 3.71e16T + 7.40e32T^{2} \)
71 \( 1 + 6.33e16iT - 2.10e33T^{2} \)
73 \( 1 + 5.51e16T + 3.46e33T^{2} \)
79 \( 1 + 7.16e16T + 1.43e34T^{2} \)
83 \( 1 - 1.18e17iT - 3.49e34T^{2} \)
89 \( 1 - 9.49e16iT - 1.22e35T^{2} \)
97 \( 1 + 3.38e17T + 5.77e35T^{2} \)
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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

−18.38647295708425014017172090512, −16.37460657873973894258115135750, −14.42333739638818187133899590036, −13.22830772240933721270975886282, −11.34434913392810402102856637120, −9.355958781691056624680816120701, −8.098142462528363529437747176132, −4.78617867487877891476828947179, −2.91397138753411590374262218766, −1.06296280784780493999662929962, 2.06752858939011897345490473136, 4.33019921875572607085754437859, 6.92455988168256863984314958708, 8.256047643991086991922305836148, 10.11560295658671357009838399503, 12.74796209308551094788426831219, 14.64300984658367303239727490826, 15.03235516249335234199423719748, 17.37079024628994687358664936901, 18.65448760198895053141335072508

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