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Properties

Label 2-1638-7.2-c1-0-38
Degree $2$
Conductor $1638$
Sign $-0.947 - 0.318i$
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 + (0.707 − 1.22i)5-s + (−1.62 − 2.09i)7-s − 0.999·8-s + (−0.707 − 1.22i)10-s + (−1.20 − 2.09i)11-s + 13-s + (−2.62 + 0.358i)14-s + (−0.5 + 0.866i)16-s + (−1.62 − 2.80i)17-s + (−1.5 + 2.59i)19-s − 1.41·20-s − 2.41·22-s + (2.53 − 4.39i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.316 − 0.547i)5-s + (−0.612 − 0.790i)7-s − 0.353·8-s + (−0.223 − 0.387i)10-s + (−0.363 − 0.630i)11-s + 0.277·13-s + (−0.700 + 0.0958i)14-s + (−0.125 + 0.216i)16-s + (−0.393 − 0.681i)17-s + (−0.344 + 0.596i)19-s − 0.316·20-s − 0.514·22-s + (0.528 − 0.915i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.947 - 0.318i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.947 - 0.318i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.118058393\)
\(L(\frac12)\) \(\approx\) \(1.118058393\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 2.09i)T \)
13 \( 1 - T \)
good5 \( 1 + (-0.707 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.20 + 2.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.62 + 2.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.53 + 4.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (0.585 + 1.01i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.53 - 7.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.41T + 41T^{2} \)
43 \( 1 + 0.828T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.15 + 8.93i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.20 - 9.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.62 - 2.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.67 - 2.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - T + 71T^{2} \)
73 \( 1 + (6.53 + 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.65 + 8.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 + (-1.12 + 1.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.990230155442161079243254037104, −8.417738030434338091985988554718, −7.21990660586359372672366292104, −6.44682306326228412609456769538, −5.49761287245705842699161019781, −4.73576611442128051539809412435, −3.73524013890505224241054850894, −2.95168054001931380070451086870, −1.61896209378890804740382336567, −0.36279140917537326071244194323, 2.03631405010554828785559619864, 2.99112764699686567252400092114, 3.96860559670531849130754250833, 5.10346289806759034771033592057, 5.81511888680953991774262337483, 6.61178060205476767957497430326, 7.18473419213227798188453818120, 8.192280651350849961362858050970, 9.033198902966712115991492246735, 9.603717602076758592938774526110

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