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Properties

Label 2-1638-273.68-c1-0-15
Degree $2$
Conductor $1638$
Sign $0.788 + 0.614i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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 + (0.377 − 0.653i)5-s + (−2.31 − 1.27i)7-s + i·8-s + (−0.653 − 0.377i)10-s + (4.23 + 2.44i)11-s + (−0.743 + 3.52i)13-s + (−1.27 + 2.31i)14-s + 16-s − 3.42·17-s + (3.15 − 1.82i)19-s + (−0.377 + 0.653i)20-s + (2.44 − 4.23i)22-s + 2.97i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.168 − 0.292i)5-s + (−0.875 − 0.482i)7-s + 0.353i·8-s + (−0.206 − 0.119i)10-s + (1.27 + 0.737i)11-s + (−0.206 + 0.978i)13-s + (−0.341 + 0.619i)14-s + 0.250·16-s − 0.831·17-s + (0.724 − 0.418i)19-s + (−0.0843 + 0.146i)20-s + (0.521 − 0.903i)22-s + 0.620i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.788 + 0.614i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.788 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.550741469\)
\(L(\frac12)\) \(\approx\) \(1.550741469\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.31 + 1.27i)T \)
13 \( 1 + (0.743 - 3.52i)T \)
good5 \( 1 + (-0.377 + 0.653i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.23 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.42T + 17T^{2} \)
19 \( 1 + (-3.15 + 1.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.97iT - 23T^{2} \)
29 \( 1 + (-2.55 + 1.47i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.86 + 3.38i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.60T + 37T^{2} \)
41 \( 1 + (1.57 + 2.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.02 - 1.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0925 + 0.160i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.09 - 2.93i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.86T + 59T^{2} \)
61 \( 1 + (-7.49 + 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.34 + 4.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.33 - 4.23i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.00 + 2.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.83 + 8.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.66T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + (5.57 + 3.21i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.512064361912033538039435867218, −8.934729127300536837752902302285, −7.67940610291730273511239190070, −6.79475897209457154380706838562, −6.22436946827424768279781804999, −4.82054614457564998871265590056, −4.22834435291330657463638190147, −3.30876641466084175372389192654, −2.11717246868613096687758464185, −0.979372848951499921648371231976, 0.811648181638396066058305243869, 2.67338406807269626076852245348, 3.48204014388777276032519132783, 4.58111798121201962342156322894, 5.62039571698812058717266452133, 6.49358647659000670467011579628, 6.64505663980083228763181979350, 7.973810324623252767639837883307, 8.606297238363130657188170749446, 9.374331324936925049315741241437

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