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Properties

Label 2-2646-63.47-c1-0-17
Degree $2$
Conductor $2646$
Sign $0.998 + 0.0552i$
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 + 0.366·5-s − 0.999i·8-s + (0.317 − 0.183i)10-s + 0.669i·11-s + (0.867 − 0.500i)13-s + (−0.5 − 0.866i)16-s + (2.49 + 4.32i)17-s + (5.50 + 3.17i)19-s + (0.183 − 0.317i)20-s + (0.334 + 0.579i)22-s + 7.69i·23-s − 4.86·25-s + (0.500 − 0.867i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.163·5-s − 0.353i·8-s + (0.100 − 0.0579i)10-s + 0.201i·11-s + (0.240 − 0.138i)13-s + (−0.125 − 0.216i)16-s + (0.605 + 1.04i)17-s + (1.26 + 0.729i)19-s + (0.0409 − 0.0709i)20-s + (0.0713 + 0.123i)22-s + 1.60i·23-s − 0.973·25-s + (0.0982 − 0.170i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.998 + 0.0552i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.998 + 0.0552i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.828732654\)
\(L(\frac12)\) \(\approx\) \(2.828732654\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.366T + 5T^{2} \)
11 \( 1 - 0.669iT - 11T^{2} \)
13 \( 1 + (-0.867 + 0.500i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.49 - 4.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.69iT - 23T^{2} \)
29 \( 1 + (1.58 + 0.914i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.47 - 3.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.58 + 4.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.15 + 3.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.24 + 3.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.16 - 7.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.36 + 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.29 - 2.47i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.49iT - 71T^{2} \)
73 \( 1 + (-3.52 + 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.17 + 7.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.50 + 14.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.35 - 9.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.9 - 8.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.015322298095191857450343714948, −7.84460541352761811798652773492, −7.48785664573890549411803434631, −6.24119515357451098887191855566, −5.72553116210161972719815473707, −5.00763211480475241122091259943, −3.80891260246610763399954546562, −3.40837113647807716816057132764, −2.10184343161515355806777110848, −1.19132500019339446674631270835, 0.871176428662131582614965526454, 2.43933262625564279090574335854, 3.15521495949705134845065750986, 4.22702187520910900529379511089, 4.98498845737110412821587851839, 5.73548679266520062705536555733, 6.52198130792508542664490505367, 7.25142720117404275434455978655, 7.984374736950246022932984879495, 8.757047758249413945025277148568

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