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Properties

Label 2-1638-273.68-c1-0-30
Degree $2$
Conductor $1638$
Sign $0.902 + 0.431i$
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 + (0.860 − 1.49i)5-s + (2.52 − 0.778i)7-s − i·8-s + (1.49 + 0.860i)10-s + (−4.41 − 2.55i)11-s + (−2.57 + 2.52i)13-s + (0.778 + 2.52i)14-s + 16-s + 4.73·17-s + (1.61 − 0.930i)19-s + (−0.860 + 1.49i)20-s + (2.55 − 4.41i)22-s + 1.24i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.384 − 0.666i)5-s + (0.955 − 0.294i)7-s − 0.353i·8-s + (0.471 + 0.272i)10-s + (−1.33 − 0.768i)11-s + (−0.713 + 0.700i)13-s + (0.207 + 0.675i)14-s + 0.250·16-s + 1.14·17-s + (0.369 − 0.213i)19-s + (−0.192 + 0.333i)20-s + (0.543 − 0.941i)22-s + 0.258i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.902 + 0.431i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.667110187\)
\(L(\frac12)\) \(\approx\) \(1.667110187\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 + 0.778i)T \)
13 \( 1 + (2.57 - 2.52i)T \)
good5 \( 1 + (-0.860 + 1.49i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.41 + 2.55i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + (-1.61 + 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.24iT - 23T^{2} \)
29 \( 1 + (-7.61 + 4.39i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.567 + 0.327i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 + (5.89 + 10.2i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.34 + 5.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.55 + 7.88i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.88 + 5.70i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.35T + 59T^{2} \)
61 \( 1 + (5.63 - 3.25i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.30 + 5.72i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.7 + 6.76i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.22 - 1.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.19 + 8.99i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 - 1.11T + 89T^{2} \)
97 \( 1 + (-8.46 - 4.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.011006395346813673027573886169, −8.529962043834812808922902901651, −7.65898300149930942564861981565, −7.17618112838217392271515043241, −5.85711921412835212743841301957, −5.21343032468380486706082470419, −4.74007894018895657388207522629, −3.46916662822990507323198343893, −2.09179186674558037591297230435, −0.69794642153828794456612734613, 1.31838362995645565288756385791, 2.59815299150557027341008160105, 2.98665826839398867107706511891, 4.61766062384490872927394300592, 5.11330579289791370958197548434, 6.00152680620287504726232213432, 7.31385065986073459446010910967, 7.85822195633946014247508544226, 8.623907282269618755071014373251, 9.798291464659850405770750024835

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