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Properties

Label 2-1638-13.10-c1-0-13
Degree $2$
Conductor $1638$
Sign $0.702 - 0.711i$
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 + i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−0.5 − 0.866i)10-s + (−0.633 + 0.366i)11-s + (2.59 + 2.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (2.86 − 4.96i)17-s + (−1.26 − 0.732i)19-s + (0.866 + 0.499i)20-s + (0.366 − 0.633i)22-s + (−0.633 − 1.09i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.158 − 0.273i)10-s + (−0.191 + 0.110i)11-s + (0.720 + 0.693i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.695 − 1.20i)17-s + (−0.290 − 0.167i)19-s + (0.193 + 0.111i)20-s + (0.0780 − 0.135i)22-s + (−0.132 − 0.228i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.702 - 0.711i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202373553\)
\(L(\frac12)\) \(\approx\) \(1.202373553\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 2.5i)T \)
good5 \( 1 - iT - 5T^{2} \)
11 \( 1 + (0.633 - 0.366i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.86 + 4.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.26 + 0.732i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.26iT - 31T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.13 + 1.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.09 - 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.92iT - 47T^{2} \)
53 \( 1 + 1.53T + 53T^{2} \)
59 \( 1 + (-9.29 - 5.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.0 + 5.83i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12 - 6.92i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 3.80T + 79T^{2} \)
83 \( 1 - 3.80iT - 83T^{2} \)
89 \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.73 + 2.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.578182061141252313136735783895, −8.660126706929293786858066849659, −7.919355930016968679889211903476, −6.97384598195482195606025219356, −6.57570885602291142428691896438, −5.55966994112828345293916444993, −4.58060573958407133701005564718, −3.39629012528577938386168377036, −2.38792041070026350431281143392, −0.938097808351293858212214649540, 0.77608710007335953795635946107, 1.98471603129008609565344045543, 3.24065522181608972633226210324, 3.97814081166606241115861380771, 5.30636370700263271083463884035, 6.03058864238415102528057388854, 6.97223674339414195859844115084, 8.087220666973329506654922339726, 8.405245967650287882132009394256, 9.263563185062782030541912576948

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