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Properties

Label 2-2646-63.20-c1-0-5
Degree $2$
Conductor $2646$
Sign $0.549 - 0.835i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.80 − 3.13i)5-s − 0.999i·8-s + 3.61i·10-s + (−1.73 − 1.00i)11-s + (−2.95 + 1.70i)13-s + (−0.5 + 0.866i)16-s + 6.17·17-s − 1.01i·19-s + (1.80 − 3.13i)20-s + (1.00 + 1.73i)22-s + (−2.62 + 1.51i)23-s + (−4.04 + 7.01i)25-s + 3.40·26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.809 − 1.40i)5-s − 0.353i·8-s + 1.14i·10-s + (−0.523 − 0.302i)11-s + (−0.818 + 0.472i)13-s + (−0.125 + 0.216i)16-s + 1.49·17-s − 0.232i·19-s + (0.404 − 0.700i)20-s + (0.213 + 0.369i)22-s + (−0.546 + 0.315i)23-s + (−0.809 + 1.40i)25-s + 0.668·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.549 - 0.835i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3369894666\)
\(L(\frac12)\) \(\approx\) \(0.3369894666\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (1.80 + 3.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.73 + 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.95 - 1.70i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.17T + 17T^{2} \)
19 \( 1 + 1.01iT - 19T^{2} \)
23 \( 1 + (2.62 - 1.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.04 + 2.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.787 - 0.454i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + (2.85 + 4.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.39 - 4.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.75iT - 53T^{2} \)
59 \( 1 + (4.49 + 7.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.7 - 7.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 7.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.466iT - 71T^{2} \)
73 \( 1 + 4.21iT - 73T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.00 - 6.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.78T + 89T^{2} \)
97 \( 1 + (-10.1 - 5.87i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.951709711345913778540452237124, −8.238480440761096237633632268403, −7.72580223105832101492269044349, −7.05804670241135425991336605788, −5.66574553275120499121531056405, −5.08020989233251963773630996918, −4.11362887408717598530170344164, −3.34905006728975446132727491409, −2.04803230193381382551396302626, −0.928982517660195677106604929136, 0.16707147291691916498909934071, 1.95779161361249170022310599478, 3.03379871121219361606630874733, 3.66873335710381779516126713313, 4.99861359068127133152772378500, 5.76136643685342576961237927447, 6.75704445229206023796492375130, 7.35311353151536952565946794770, 7.80628094797912772028849548786, 8.481707228924403261394169970415

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