Properties

Label 2-1638-273.62-c1-0-39
Degree $2$
Conductor $1638$
Sign $-0.915 + 0.401i$
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 − 4.13i·5-s + (−2.07 + 1.64i)7-s − 0.999·8-s + (3.57 − 2.06i)10-s + (−1.97 − 3.41i)11-s + (3.54 − 0.675i)13-s + (−2.45 − 0.976i)14-s + (−0.5 − 0.866i)16-s + (2.00 − 3.47i)17-s + (−4.16 + 7.21i)19-s + (3.57 + 2.06i)20-s + (1.97 − 3.41i)22-s + (−5.53 + 3.19i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.84i·5-s + (−0.784 + 0.620i)7-s − 0.353·8-s + (1.13 − 0.653i)10-s + (−0.594 − 1.02i)11-s + (0.982 − 0.187i)13-s + (−0.657 − 0.260i)14-s + (−0.125 − 0.216i)16-s + (0.487 − 0.843i)17-s + (−0.956 + 1.65i)19-s + (0.799 + 0.461i)20-s + (0.420 − 0.727i)22-s + (−1.15 + 0.665i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.915 + 0.401i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.915 + 0.401i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3137196091\)
\(L(\frac12)\) \(\approx\) \(0.3137196091\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 - 1.64i)T \)
13 \( 1 + (-3.54 + 0.675i)T \)
good5 \( 1 + 4.13iT - 5T^{2} \)
11 \( 1 + (1.97 + 3.41i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.00 + 3.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.16 - 7.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.53 - 3.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.10 - 3.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.23T + 31T^{2} \)
37 \( 1 + (5.40 - 3.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.96 - 1.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.59 + 4.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.33iT - 47T^{2} \)
53 \( 1 + 0.335iT - 53T^{2} \)
59 \( 1 + (-1.89 - 1.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.2 + 5.93i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.8 - 7.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 + 9.00i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 + 3.93iT - 83T^{2} \)
89 \( 1 + (7.32 - 4.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.21 + 5.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.788309783176977972503389005598, −8.328836885814606210482317246306, −7.69981984011364181378581023167, −6.19556168747861965351896628106, −5.74758359770279810247990175627, −5.18489675897406226670178668006, −4.01918189175372786771441878428, −3.29126690988140062381739325377, −1.61591918049763317380073582322, −0.10070584083662578953057341733, 2.02772125652149634941885561869, 2.81042130748018186423977233415, 3.73276599383367799285915353410, 4.36387484523181230989885144539, 5.89995177644902832252319589664, 6.49234617073528639620438326815, 7.12672862749608133797417833673, 7.996117641511536070879653463091, 9.234910281656769110155303295115, 10.10413734843542846482387443428

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