Properties

Label 2-6-3.2-c16-0-1
Degree $2$
Conductor $6$
Sign $0.336 - 0.941i$
Analytic cond. $9.73947$
Root an. cond. $3.12081$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 181. i·2-s + (6.17e3 + 2.20e3i)3-s − 3.27e4·4-s + 5.48e5i·5-s + (3.99e5 − 1.11e6i)6-s − 8.81e6·7-s + 5.93e6i·8-s + (3.32e7 + 2.72e7i)9-s + 9.92e7·10-s + 2.97e8i·11-s + (−2.02e8 − 7.23e7i)12-s + 6.77e8·13-s + 1.59e9i·14-s + (−1.21e9 + 3.38e9i)15-s + 1.07e9·16-s − 3.75e9i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.941 + 0.336i)3-s − 0.500·4-s + 1.40i·5-s + (0.238 − 0.665i)6-s − 1.52·7-s + 0.353i·8-s + (0.773 + 0.634i)9-s + 0.992·10-s + 1.38i·11-s + (−0.470 − 0.168i)12-s + 0.830·13-s + 1.08i·14-s + (−0.472 + 1.32i)15-s + 0.250·16-s − 0.538i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(9.73947\)
Root analytic conductor: \(3.12081\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :8),\ 0.336 - 0.941i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.38795 + 0.977760i\)
\(L(\frac12)\) \(\approx\) \(1.38795 + 0.977760i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 181. iT \)
3 \( 1 + (-6.17e3 - 2.20e3i)T \)
good5 \( 1 - 5.48e5iT - 1.52e11T^{2} \)
7 \( 1 + 8.81e6T + 3.32e13T^{2} \)
11 \( 1 - 2.97e8iT - 4.59e16T^{2} \)
13 \( 1 - 6.77e8T + 6.65e17T^{2} \)
17 \( 1 + 3.75e9iT - 4.86e19T^{2} \)
19 \( 1 + 9.70e8T + 2.88e20T^{2} \)
23 \( 1 + 2.68e10iT - 6.13e21T^{2} \)
29 \( 1 - 3.75e11iT - 2.50e23T^{2} \)
31 \( 1 + 4.78e11T + 7.27e23T^{2} \)
37 \( 1 - 9.79e11T + 1.23e25T^{2} \)
41 \( 1 + 1.07e13iT - 6.37e25T^{2} \)
43 \( 1 - 8.94e12T + 1.36e26T^{2} \)
47 \( 1 - 2.80e13iT - 5.66e26T^{2} \)
53 \( 1 + 7.34e13iT - 3.87e27T^{2} \)
59 \( 1 - 1.35e14iT - 2.15e28T^{2} \)
61 \( 1 - 4.44e13T + 3.67e28T^{2} \)
67 \( 1 - 6.70e14T + 1.64e29T^{2} \)
71 \( 1 - 6.75e14iT - 4.16e29T^{2} \)
73 \( 1 - 4.82e14T + 6.50e29T^{2} \)
79 \( 1 - 3.05e14T + 2.30e30T^{2} \)
83 \( 1 + 1.62e15iT - 5.07e30T^{2} \)
89 \( 1 - 3.89e15iT - 1.54e31T^{2} \)
97 \( 1 - 1.41e16T + 6.14e31T^{2} \)
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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

−19.32878555191323513607667324831, −18.32050377136498189440019602331, −15.74917290751665689999918988256, −14.35699216036824260266236093852, −12.86408880357099527148798080929, −10.53725329518493237302244383137, −9.450931552343509865561183915468, −6.99688512224407447495996652940, −3.66337044170614071331562990011, −2.49031371516011271071657554530, 0.74879565564076440136694501021, 3.63165854064562074591928381867, 6.13930717651991674205615488518, 8.316903820859656618193592963459, 9.353985126093516907070868432293, 12.81505850937525346743821105561, 13.59127463853632432735875182465, 15.73428745662231062758418438088, 16.64234089813135749250176796230, 18.82310474221468497564362417951

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