Properties

Label 2-2646-63.58-c1-0-28
Degree $2$
Conductor $2646$
Sign $-0.154 + 0.987i$
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  + 2-s + 4-s + (−0.794 − 1.37i)5-s + 8-s + (−0.794 − 1.37i)10-s + (−0.794 + 1.37i)11-s + (−2.40 + 4.16i)13-s + 16-s + (−2.69 − 4.67i)17-s + (3.54 − 6.14i)19-s + (−0.794 − 1.37i)20-s + (−0.794 + 1.37i)22-s + (0.150 + 0.260i)23-s + (1.23 − 2.14i)25-s + (−2.40 + 4.16i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.355 − 0.615i)5-s + 0.353·8-s + (−0.251 − 0.434i)10-s + (−0.239 + 0.414i)11-s + (−0.667 + 1.15i)13-s + 0.250·16-s + (−0.654 − 1.13i)17-s + (0.814 − 1.41i)19-s + (−0.177 − 0.307i)20-s + (−0.169 + 0.293i)22-s + (0.0313 + 0.0542i)23-s + (0.247 − 0.429i)25-s + (−0.471 + 0.817i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.154 + 0.987i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959310781\)
\(L(\frac12)\) \(\approx\) \(1.959310781\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.794 - 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.40 - 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.150 - 0.260i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.71T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (8.02 + 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.712 + 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.793767359598425702965963708698, −7.57733240086590995935531293260, −7.18668953712356965599026889407, −6.36527757898199547279125141536, −5.25098212574126370373068025264, −4.66081946104972474058669399303, −4.15035304989601197833022837080, −2.82907377672519566517397395772, −2.08489891374438329445174584734, −0.48954157399292824475978867295, 1.42820842158231725149751660901, 2.77411492085323730793495917982, 3.36658373412210745385703383236, 4.20593193592584605247715076513, 5.31109376289310881764007882691, 5.82736829883518280664212743225, 6.72322618717048950595320867380, 7.56600303669821737611223798748, 8.025875713958985525182077890056, 8.988625942720993642611434570366

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