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Properties

Label 2-1638-273.68-c1-0-10
Degree $2$
Conductor $1638$
Sign $-0.803 - 0.595i$
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  + i·2-s − 4-s + (0.367 − 0.636i)5-s + (2.16 + 1.51i)7-s − i·8-s + (0.636 + 0.367i)10-s + (1.23 + 0.713i)11-s + (−3.48 + 0.941i)13-s + (−1.51 + 2.16i)14-s + 16-s − 4.00·17-s + (−4.71 + 2.72i)19-s + (−0.367 + 0.636i)20-s + (−0.713 + 1.23i)22-s + 1.14i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (0.164 − 0.284i)5-s + (0.819 + 0.573i)7-s − 0.353i·8-s + (0.201 + 0.116i)10-s + (0.372 + 0.215i)11-s + (−0.965 + 0.260i)13-s + (−0.405 + 0.579i)14-s + 0.250·16-s − 0.972·17-s + (−1.08 + 0.624i)19-s + (−0.0821 + 0.142i)20-s + (−0.152 + 0.263i)22-s + 0.239i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.280839434\)
\(L(\frac12)\) \(\approx\) \(1.280839434\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.16 - 1.51i)T \)
13 \( 1 + (3.48 - 0.941i)T \)
good5 \( 1 + (-0.367 + 0.636i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.23 - 0.713i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 + (4.71 - 2.72i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.14iT - 23T^{2} \)
29 \( 1 + (-2.63 + 1.52i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.05 - 1.18i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.84T + 37T^{2} \)
41 \( 1 + (-4.44 - 7.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.70 - 9.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.243 - 0.422i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.7 + 6.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.37T + 59T^{2} \)
61 \( 1 + (3.51 - 2.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.212 - 0.367i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.3 + 7.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (14.0 - 8.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.52 - 4.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.00T + 83T^{2} \)
89 \( 1 + 9.89T + 89T^{2} \)
97 \( 1 + (-10.3 - 5.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.463166830496999066289243510899, −8.792739308788346365241014451964, −8.157368982466765744361008493538, −7.30980532843075428306651295154, −6.47726480038284589369398263547, −5.66785600929355756922342489019, −4.73673551609563603488299147721, −4.25268946566986282519262378941, −2.64247462793317797139926427560, −1.53008422094594218213245375316, 0.48770171628426415411964629401, 1.98250076281592534591772248745, 2.74727654513342849157169279784, 4.14885634360594333114350372088, 4.58502848161834808842313111338, 5.66905724368622861169532032247, 6.79468957138182423473407648172, 7.43029005817624183199547402474, 8.593580012584796203286200865537, 8.926578815844897081362610610119

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