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Properties

Label 2-2646-63.38-c1-0-21
Degree $2$
Conductor $2646$
Sign $0.881 - 0.472i$
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  + i·2-s − 4-s + (0.895 − 1.55i)5-s − i·8-s + (1.55 + 0.895i)10-s + (−2.07 + 1.20i)11-s + (−4.23 + 2.44i)13-s + 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 + (3.26 + 1.88i)23-s + (0.897 + 1.55i)25-s + (−2.44 − 4.23i)26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.400 − 0.693i)5-s − 0.353i·8-s + (0.490 + 0.283i)10-s + (−0.627 + 0.362i)11-s + (−1.17 + 0.678i)13-s + 0.250·16-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 + (0.680 + 0.392i)23-s + (0.179 + 0.310i)25-s + (−0.479 − 0.830i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.881 - 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.669284217\)
\(L(\frac12)\) \(\approx\) \(1.669284217\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.895 + 1.55i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
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 \( 1 + (-3.26 - 1.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.68 - 3.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.64iT - 31T^{2} \)
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 \( 1 - 5.13T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 - 0.570T + 67T^{2} \)
71 \( 1 + 5.96iT - 71T^{2} \)
73 \( 1 + (-10.7 - 6.19i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.03T + 79T^{2} \)
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

−9.020409334518214038164770136160, −8.037014068940902220767661686093, −7.32888553283291436516399067470, −6.82132077577296152686073668938, −5.66849321552251730005792236387, −5.00983386787349523150617670478, −4.62359150951385961223228541620, −3.24749133780365658749562860701, −2.16883451548541609590890138078, −0.77866671718357688500778824391, 0.843499690775825269203086876487, 2.30342712580042451398707244914, 2.87866966875746748201227218686, 3.74688887170898784617329318484, 4.97487544985666281630761843396, 5.49069433604725976011541034190, 6.51377039001115756930178062344, 7.31815593264568781554768290578, 8.170376265680319268990918968799, 8.798522221042892631148114057358

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