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Properties

Label 2-2646-63.47-c1-0-4
Degree $2$
Conductor $2646$
Sign $-0.0945 - 0.995i$
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 − 1.79·5-s − 0.999i·8-s + (−1.55 + 0.895i)10-s − 2.40i·11-s + (−4.23 + 2.44i)13-s + (−0.5 − 0.866i)16-s + (1.83 + 3.17i)17-s + (−2.61 − 1.50i)19-s + (−0.895 + 1.55i)20-s + (−1.20 − 2.07i)22-s − 3.76i·23-s − 1.79·25-s + (−2.44 + 4.23i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.800·5-s − 0.353i·8-s + (−0.490 + 0.283i)10-s − 0.724i·11-s + (−1.17 + 0.678i)13-s + (−0.125 − 0.216i)16-s + (0.444 + 0.769i)17-s + (−0.599 − 0.346i)19-s + (−0.200 + 0.346i)20-s + (−0.256 − 0.443i)22-s − 0.785i·23-s − 0.358·25-s + (−0.479 + 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0945 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0945 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0945 - 0.995i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.0945 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7993240670\)
\(L(\frac12)\) \(\approx\) \(0.7993240670\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.79T + 5T^{2} \)
11 \( 1 + 2.40iT - 11T^{2} \)
13 \( 1 + (4.23 - 2.44i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.83 - 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.61 + 1.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.76iT - 23T^{2} \)
29 \( 1 + (-5.68 - 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.02 - 2.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.68 - 8.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.04 - 6.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.48 - 6.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.56 + 4.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.29 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.81 - 5.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.285 - 0.493i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + (10.7 - 6.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.51 + 2.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.00 + 12.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.87 - 3.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.77 - 2.75i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.957436264267140626985169765139, −8.312092725230417346726064098465, −7.52409497317978586411934558937, −6.61517116557929488394418239264, −6.04819271710260699540024261097, −4.75231569357356807497172557178, −4.49083522626835495251956625417, −3.35912793821321478987962508288, −2.66386803608930682493971850433, −1.34212291178345598871233180110, 0.20853399635885768277014939534, 2.07774768392743324444756287513, 3.05342754210822169519432325516, 3.97205511527245312887870136572, 4.72688872834336376942460894130, 5.41131150428241683805516476502, 6.34980193685692242196891539998, 7.39756834160607190879610973170, 7.56725798652159685815465201851, 8.391025111013493288218866434053

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