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Properties

Label 2-1638-7.2-c1-0-17
Degree $2$
Conductor $1638$
Sign $0.605 - 0.795i$
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 + (2.5 − 0.866i)7-s + 0.999·8-s + (0.5 + 0.866i)11-s − 13-s + (−0.500 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−2.5 + 4.33i)19-s − 0.999·22-s + (1 − 1.73i)23-s + (2.5 + 4.33i)25-s + (0.5 − 0.866i)26-s + (−1.99 − 1.73i)28-s + 5·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.944 − 0.327i)7-s + 0.353·8-s + (0.150 + 0.261i)11-s − 0.277·13-s + (−0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.121 − 0.210i)17-s + (−0.573 + 0.993i)19-s − 0.213·22-s + (0.208 − 0.361i)23-s + (0.5 + 0.866i)25-s + (0.0980 − 0.169i)26-s + (−0.377 − 0.327i)28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.527102648\)
\(L(\frac12)\) \(\approx\) \(1.527102648\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
13 \( 1 + T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1 + 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-5.5 + 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (8 - 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.324212620875756592404289019577, −8.569669569296550001837164432766, −7.951759678861771154663020116901, −7.13686611145762101365056805473, −6.47058107421488531061756114574, −5.34779681788081531562185630051, −4.72936903056450996255081344261, −3.75075434047520526625475754431, −2.24197978126696964981295894954, −1.05437542937507547113199125490, 0.855803606293805256297140439447, 2.15243148971300611506108088343, 2.96314337780601558094647901632, 4.34744571534578184053435646016, 4.83946770356458674943453839329, 6.05151223945454954642411532183, 6.93811517446882385959504478452, 8.052407075432851372862965641755, 8.416638518301415063244384139027, 9.290163571141235009345698553588

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