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Properties

Label 2-1638-91.16-c1-0-12
Degree $2$
Conductor $1638$
Sign $0.766 - 0.642i$
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  − 2-s + 4-s + (−2.05 + 3.56i)5-s + (−2.61 − 0.405i)7-s − 8-s + (2.05 − 3.56i)10-s + (2.02 − 3.50i)11-s + (1.81 − 3.11i)13-s + (2.61 + 0.405i)14-s + 16-s − 0.715·17-s + (1.92 + 3.33i)19-s + (−2.05 + 3.56i)20-s + (−2.02 + 3.50i)22-s + 4.09·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.920 + 1.59i)5-s + (−0.988 − 0.153i)7-s − 0.353·8-s + (0.650 − 1.12i)10-s + (0.609 − 1.05i)11-s + (0.503 − 0.864i)13-s + (0.698 + 0.108i)14-s + 0.250·16-s − 0.173·17-s + (0.441 + 0.764i)19-s + (−0.460 + 0.796i)20-s + (−0.431 + 0.746i)22-s + 0.854·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8364500409\)
\(L(\frac12)\) \(\approx\) \(0.8364500409\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.405i)T \)
13 \( 1 + (-1.81 + 3.11i)T \)
good5 \( 1 + (2.05 - 3.56i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.02 + 3.50i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.715T + 17T^{2} \)
19 \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 + (4.50 + 7.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.82 - 3.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.19T + 37T^{2} \)
41 \( 1 + (-2.88 - 4.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.28 + 2.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.28 - 2.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.35 - 2.35i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + (-4.71 - 8.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.583 + 1.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.10 - 8.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.25 + 2.17i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.590165899865978344780484951266, −8.601734453643114452131329599797, −7.77321402926671015152198419916, −7.24751511141492683362744008515, −6.23288052611920574100140516036, −6.00606257449477112189619606872, −4.01013051909118290983730402465, −3.29291029319103548485284626704, −2.74035423798191915412454769017, −0.75242563158121427054608501350, 0.66500642843144444556355764730, 1.80046979761412594804723960718, 3.35920907239434093819727246486, 4.29304817552260928301353340055, 5.04169097949422544825830478996, 6.24279620617545845286497558806, 7.11451440311504435464029649468, 7.71703489996302662772503004216, 8.912366566954177016580297674485, 9.100130871211293517989158264524

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