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Properties

Label 2-2646-63.20-c1-0-1
Degree $2$
Conductor $2646$
Sign $-0.999 - 0.0207i$
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 + (−0.724 − 1.25i)5-s + 0.999i·8-s − 1.44i·10-s + (1.21 + 0.698i)11-s + (−3.03 + 1.75i)13-s + (−0.5 + 0.866i)16-s − 7.90·17-s + 4.16i·19-s + (0.724 − 1.25i)20-s + (0.698 + 1.21i)22-s + (3.13 − 1.80i)23-s + (1.45 − 2.51i)25-s − 3.50·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.324 − 0.561i)5-s + 0.353i·8-s − 0.458i·10-s + (0.364 + 0.210i)11-s + (−0.841 + 0.485i)13-s + (−0.125 + 0.216i)16-s − 1.91·17-s + 0.956i·19-s + (0.162 − 0.280i)20-s + (0.149 + 0.258i)22-s + (0.653 − 0.377i)23-s + (0.290 − 0.502i)25-s − 0.687·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.999 - 0.0207i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.999 - 0.0207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5375335790\)
\(L(\frac12)\) \(\approx\) \(0.5375335790\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.724 + 1.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.21 - 0.698i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.03 - 1.75i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.90T + 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
23 \( 1 + (-3.13 + 1.80i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.06 + 2.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.794 + 0.458i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.28T + 37T^{2} \)
41 \( 1 + (-0.343 - 0.595i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.01 - 10.4i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.15 - 7.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.96iT - 53T^{2} \)
59 \( 1 + (4.72 + 8.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.53 + 4.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.48 - 2.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + (7.81 - 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.11 - 7.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 1.06T + 89T^{2} \)
97 \( 1 + (-10.9 - 6.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.136638828635134676463353569752, −8.370201802910655685857069206624, −7.71051179492694242350005025388, −6.70823575987745381451191835727, −6.37354232765196779998019200309, −5.10763978461097553323072635173, −4.57599654846966834730588353757, −3.92705668351204747832146921308, −2.71832105399155872888756114942, −1.69858122790169411996960438497, 0.12881289527867839508963688681, 1.81780030575095145397808378186, 2.80918240472111205621385784298, 3.51659355027208432060560448694, 4.55015667907237580226604778310, 5.15221012126460792953000651834, 6.16800346022517538046279046328, 7.13698064337856125814421247600, 7.23868399682757031183606428684, 8.764236694172312249398184112408

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