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Properties

Label 2-1638-13.9-c1-0-22
Degree $2$
Conductor $1638$
Sign $0.160 + 0.987i$
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 + 2·5-s + (0.5 + 0.866i)7-s − 0.999·8-s + (1 − 1.73i)10-s + (3.08 − 5.35i)11-s + (−1.5 − 3.27i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + (4.08 + 7.08i)19-s + (−0.999 − 1.73i)20-s + (−3.08 − 5.35i)22-s + (1.58 − 2.75i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.894·5-s + (0.188 + 0.327i)7-s − 0.353·8-s + (0.316 − 0.547i)10-s + (0.931 − 1.61i)11-s + (−0.416 − 0.909i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + (0.938 + 1.62i)19-s + (−0.223 − 0.387i)20-s + (−0.658 − 1.14i)22-s + (0.331 − 0.574i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.160 + 0.987i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.542778389\)
\(L(\frac12)\) \(\approx\) \(2.542778389\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 3.27i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + (-3.08 + 5.35i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.08 - 7.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.58 + 2.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.08 + 7.08i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.17T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.08 + 3.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.410 + 0.711i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.17T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (0.410 + 0.711i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.58 + 13.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.58 - 7.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 - 5.17T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.17 + 10.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.400239561059035650880596083765, −8.464827383927679632418152897626, −7.86820659430120562280874881479, −6.35880283920093600813665010317, −5.80341522257816746191633106933, −5.32860985887776815774613597530, −3.86027381375004908146314104389, −3.24309705286903968104694965430, −2.03659364531664558524411816611, −0.987768247404151577080456668725, 1.41703385047007484129623333474, 2.55890970485617634910853016425, 3.82872327931001945222784813670, 4.94377072181008046470830919125, 5.18016354455645884575561538767, 6.60540996448269782236696571406, 7.03526413709336804130185492535, 7.58147614722459663468050950077, 9.113050578918500956432341932248, 9.355042327861222687647880719963

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