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Properties

Label 2-1638-91.81-c1-0-26
Degree $2$
Conductor $1638$
Sign $0.841 + 0.540i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.114 − 0.197i)5-s + (0.848 + 2.50i)7-s + 0.999·8-s + 0.228·10-s − 3.41·11-s + (−1.62 − 3.21i)13-s + (−2.59 − 0.518i)14-s + (−0.5 + 0.866i)16-s + (−2.70 − 4.68i)17-s + 6.34·19-s + (−0.114 + 0.197i)20-s + (1.70 − 2.95i)22-s + (−0.959 + 1.66i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.0509 − 0.0883i)5-s + (0.320 + 0.947i)7-s + 0.353·8-s + 0.0721·10-s − 1.02·11-s + (−0.451 − 0.892i)13-s + (−0.693 − 0.138i)14-s + (−0.125 + 0.216i)16-s + (−0.656 − 1.13i)17-s + 1.45·19-s + (−0.0254 + 0.0441i)20-s + (0.363 − 0.629i)22-s + (−0.200 + 0.346i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.841 + 0.540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.000262083\)
\(L(\frac12)\) \(\approx\) \(1.000262083\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.848 - 2.50i)T \)
13 \( 1 + (1.62 + 3.21i)T \)
good5 \( 1 + (0.114 + 0.197i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.41T + 11T^{2} \)
17 \( 1 + (2.70 + 4.68i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
23 \( 1 + (0.959 - 1.66i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.851 + 1.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.78 - 3.08i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 + 3.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.59 + 2.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.17 + 8.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.57 - 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.31 + 5.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.38 + 12.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.71T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + (-0.0390 + 0.0677i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.31 - 10.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.811 - 1.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.70T + 83T^{2} \)
89 \( 1 + (-8.77 + 15.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.82 + 11.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.217784911748752698317409184105, −8.470809799442655154000755240525, −7.66757571895651029132475300526, −7.18252753989974859233355517651, −5.90602793165748669806784491703, −5.31432332223546737617740440774, −4.71580945190259000109373531381, −3.10835695716242215032755658929, −2.23436721857784717619854994505, −0.48276349837277409862688204566, 1.16249341399513119966332406450, 2.31300011580985983808233339106, 3.42466563795809102809934843750, 4.34635147295406694466616699000, 5.11267602229421602142717304510, 6.33367471408955660820185512470, 7.43532942840873764268285472081, 7.71620897064085275213509681478, 8.799226605026418047740200534722, 9.519110331016310001412576027484

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