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Properties

Label 2-1368-19.7-c1-0-0
Degree $2$
Conductor $1368$
Sign $-0.350 - 0.936i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (−1.61 − 2.80i)5-s − 0.236·7-s − 1.23·11-s + (−1.73 + 3.00i)13-s + (−2 − 3.46i)17-s + (−2 + 3.87i)19-s + (−1.61 + 2.80i)23-s + (−2.73 + 4.73i)25-s + (−0.763 + 1.32i)29-s + 4.70·31-s + (0.381 + 0.661i)35-s + 7·37-s + (−1.23 − 2.14i)41-s + (3.11 + 5.40i)43-s + (−1 + 1.73i)47-s + ⋯
L(s)  = 1  + (−0.723 − 1.25i)5-s − 0.0892·7-s − 0.372·11-s + (−0.481 + 0.833i)13-s + (−0.485 − 0.840i)17-s + (−0.458 + 0.888i)19-s + (−0.337 + 0.584i)23-s + (−0.547 + 0.947i)25-s + (−0.141 + 0.245i)29-s + 0.845·31-s + (0.0645 + 0.111i)35-s + 1.15·37-s + (−0.193 − 0.334i)41-s + (0.475 + 0.823i)43-s + (−0.145 + 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.350 - 0.936i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3931715458\)
\(L(\frac12)\) \(\approx\) \(0.3931715458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (2 - 3.87i)T \)
good5 \( 1 + (1.61 + 2.80i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.236T + 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + (1.73 - 3.00i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (1.61 - 2.80i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.763 - 1.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (1.23 + 2.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.11 - 5.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.09 - 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.85 - 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.118 - 0.204i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.70 - 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.73 + 3.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.35 + 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-1.85 + 3.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.47 + 9.47i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.560736246633347638635463101404, −9.118927925816652795634587648658, −8.139415160849801499599902898719, −7.65663440351475935793946444037, −6.59059120905806668986361648132, −5.55768567857125020112315371487, −4.59413723890243525634015887530, −4.13944126816914486602497196790, −2.72837358833191840605934335438, −1.34064148938937775250892871359, 0.16399107698794566077206727625, 2.31472616283078355505843554962, 3.10168009310015880029723638675, 4.07829304413236722299300433479, 5.07103146679759636000309808550, 6.34029187099674153878400466685, 6.77574757742701445647246781948, 7.87395246181889550110261210881, 8.201844507516945066973359715585, 9.476870963751480397123119727914

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