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Properties

Label 2-1638-13.10-c1-0-4
Degree $2$
Conductor $1638$
Sign $-0.967 - 0.252i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 2.73i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−1.36 − 2.36i)10-s + (−0.232 + 0.133i)11-s + (0.866 − 3.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (2.13 + 1.23i)19-s + (2.36 + 1.36i)20-s + (0.133 − 0.232i)22-s + (1.73 + 3i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 1.22i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.431 − 0.748i)10-s + (−0.0699 + 0.0403i)11-s + (0.240 − 0.970i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.452 + 0.783i)17-s + (0.489 + 0.282i)19-s + (0.529 + 0.305i)20-s + (0.0285 − 0.0494i)22-s + (0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.967 - 0.252i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.967 - 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6739068824\)
\(L(\frac12)\) \(\approx\) \(0.6739068824\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 3.5i)T \)
good5 \( 1 - 2.73iT - 5T^{2} \)
11 \( 1 + (0.232 - 0.133i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.13 - 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.66iT - 31T^{2} \)
37 \( 1 + (4.09 - 2.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.06 - 3.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.36 - 2.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.46iT - 47T^{2} \)
53 \( 1 - 3.53T + 53T^{2} \)
59 \( 1 + (11.1 + 6.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.803 - 0.464i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.09 + 4.09i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (0.401 - 0.232i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.36 - 1.36i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.925447881416853158443715300023, −8.879409914138211250412094832541, −8.094468921166603597454648357054, −7.32273689053309765503682680871, −6.66034027902682380566389683399, −5.98056273394460709791438895487, −5.00341462446679008297272257898, −3.53604627461980596669166654084, −2.91509341430353355941706197240, −1.50015895838871874973977241559, 0.32238785450571022440084408146, 1.58045620115064456489568869081, 2.69507613829061416143134027523, 3.95922014038987569992646681516, 4.78198498924502143238538609453, 5.70332936247945178580541172926, 6.79886616170387734641577451601, 7.49941275384340288358580455041, 8.661575586926538717125580016523, 8.933077887236577105962300239247

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