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Properties

Label 2-1638-273.68-c1-0-6
Degree $2$
Conductor $1638$
Sign $-0.866 - 0.499i$
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 + (1.60 − 2.78i)5-s + (−2.28 + 1.34i)7-s − i·8-s + (2.78 + 1.60i)10-s + (−1.16 − 0.671i)11-s + (−3.22 + 1.62i)13-s + (−1.34 − 2.28i)14-s + 16-s + 1.49·17-s + (−2.36 + 1.36i)19-s + (−1.60 + 2.78i)20-s + (0.671 − 1.16i)22-s + 6.73i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.719 − 1.24i)5-s + (−0.862 + 0.506i)7-s − 0.353i·8-s + (0.881 + 0.508i)10-s + (−0.350 − 0.202i)11-s + (−0.893 + 0.449i)13-s + (−0.358 − 0.609i)14-s + 0.250·16-s + 0.361·17-s + (−0.541 + 0.312i)19-s + (−0.359 + 0.623i)20-s + (0.143 − 0.247i)22-s + 1.40i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.866 - 0.499i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7226661637\)
\(L(\frac12)\) \(\approx\) \(0.7226661637\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.28 - 1.34i)T \)
13 \( 1 + (3.22 - 1.62i)T \)
good5 \( 1 + (-1.60 + 2.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.16 + 0.671i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.49T + 17T^{2} \)
19 \( 1 + (2.36 - 1.36i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.73iT - 23T^{2} \)
29 \( 1 + (3.42 - 1.97i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.51 - 0.874i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.43T + 37T^{2} \)
41 \( 1 + (-1.92 - 3.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.103 + 0.179i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.95 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.08 - 4.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.70T + 59T^{2} \)
61 \( 1 + (4.80 - 2.77i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.58 - 6.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.2 - 5.90i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.20 - 1.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.259 + 0.450i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 - 2.49T + 89T^{2} \)
97 \( 1 + (12.6 + 7.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.518983797920932900773994368846, −9.044953923286721761024960404092, −8.146241509924032573832680045091, −7.36077970586525059547809241188, −6.30645529345299230087415684482, −5.64518771630607522975171080953, −5.06087741437752372251796288229, −4.06236024671755268106427195692, −2.76386518135469720270111289370, −1.42548082200563698667151583318, 0.26949524426314835476390006793, 2.18991053404805401215494354848, 2.78243750048127756364132723850, 3.68588240511236901844977202954, 4.79002028221415072426876911977, 5.88186306294333383799851638199, 6.64355669180163322687766008805, 7.33787093900639598317247233603, 8.296145340834512639499124112649, 9.592416926381942518810009938177

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