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Properties

Label 2-1638-13.9-c1-0-26
Degree $2$
Conductor $1638$
Sign $-0.396 + 0.918i$
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 + 2.56·5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (1.28 − 2.21i)10-s + (0.780 − 1.35i)11-s + (−0.5 + 3.57i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−4.06 − 7.03i)17-s + (−0.780 − 1.35i)19-s + (−1.28 − 2.21i)20-s + (−0.780 − 1.35i)22-s + (3.56 − 6.16i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.14·5-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (0.405 − 0.701i)10-s + (0.235 − 0.407i)11-s + (−0.138 + 0.990i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.985 − 1.70i)17-s + (−0.179 − 0.310i)19-s + (−0.286 − 0.496i)20-s + (−0.166 − 0.288i)22-s + (0.742 − 1.28i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.190727012\)
\(L(\frac12)\) \(\approx\) \(2.190727012\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 3.57i)T \)
good5 \( 1 - 2.56T + 5T^{2} \)
11 \( 1 + (-0.780 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.06 + 7.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.780 + 1.35i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.56 + 6.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.06 + 1.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-3.28 + 5.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.62 + 4.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + (1.56 + 2.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.62 + 4.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.438 - 0.759i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.68 - 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 + 2.43T + 79T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + (-3.78 + 6.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.56 - 7.90i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.272077656292792732006851573393, −8.790508363165006856458910451550, −7.30287078457626782251921494659, −6.58093450482566888927716728019, −5.86765774386238263638022684472, −4.81514980347337709897850752316, −4.19742672576276954068705998488, −2.77191939724509779611945423332, −2.18853298098297566924413809656, −0.74637685495659291268656313059, 1.58731383008501234063753043307, 2.69501353165692838753041901982, 3.82207054287793090124277997255, 4.86465280695480611237745380927, 5.83291960401309929747625185446, 6.11312163058295624448051015148, 7.11328192117832148569783994402, 8.004467550132142299670602782496, 8.859937899057456836695482883359, 9.487793828799615733797129524054

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