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Properties

Label 2-2646-63.20-c1-0-0
Degree $2$
Conductor $2646$
Sign $0.301 - 0.953i$
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 + (−0.995 − 1.72i)5-s − 0.999i·8-s + 1.99i·10-s + (−5.21 − 3.00i)11-s + (−3.43 + 1.98i)13-s + (−0.5 + 0.866i)16-s − 1.56·17-s − 4.80i·19-s + (0.995 − 1.72i)20-s + (3.00 + 5.21i)22-s + (5.02 − 2.90i)23-s + (0.519 − 0.899i)25-s + 3.96·26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.445 − 0.770i)5-s − 0.353i·8-s + 0.629i·10-s + (−1.57 − 0.907i)11-s + (−0.951 + 0.549i)13-s + (−0.125 + 0.216i)16-s − 0.378·17-s − 1.10i·19-s + (0.222 − 0.385i)20-s + (0.641 + 1.11i)22-s + (1.04 − 0.604i)23-s + (0.103 − 0.179i)25-s + 0.777·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.301 - 0.953i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2155555163\)
\(L(\frac12)\) \(\approx\) \(0.2155555163\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.995 + 1.72i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.21 + 3.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.43 - 1.98i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 + (-5.02 + 2.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.26 + 3.03i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.11 - 0.643i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.05T + 37T^{2} \)
41 \( 1 + (-2.98 - 5.16i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.53 - 7.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.39 - 4.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.39iT - 53T^{2} \)
59 \( 1 + (-6.38 - 11.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.61 - 0.933i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.79 + 6.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.48iT - 71T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + (-6.94 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.35 + 5.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + (14.1 + 8.14i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.052570617671219578612254929178, −8.275193485411471297289714318312, −7.73795158347731484375033589995, −6.97325192346973120321913873102, −5.92769185270795798424901832385, −4.87298482330472769517564322554, −4.43507823648198196696606263964, −2.99003549518396211716251148771, −2.43490925177918148445552266838, −0.877863407193608830098095401661, 0.10810404202795911139170531887, 1.94367203060954229576523345955, 2.78813482944261925079505326486, 3.78392078510508212703211334295, 5.16040085196230191206002232081, 5.42808924957161561206999581175, 6.79726952428002827685634921924, 7.30421433632077311125664517863, 7.77940499562124547070443487295, 8.526070347417691827427326993773

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