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Properties

Label 2-1368-19.11-c1-0-12
Degree $2$
Conductor $1368$
Sign $0.910 + 0.412i$
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  + (1.5 − 2.59i)5-s + 4·11-s + (2.5 + 4.33i)13-s + (−2.5 + 4.33i)17-s + (4 − 1.73i)19-s + (−0.5 − 0.866i)23-s + (−2 − 3.46i)25-s + (1.5 + 2.59i)29-s + 4·31-s + 2·37-s + (−2.5 + 4.33i)41-s + (5.5 − 9.52i)43-s + (−2.5 − 4.33i)47-s − 7·49-s + (−4.5 − 7.79i)53-s + ⋯
L(s)  = 1  + (0.670 − 1.16i)5-s + 1.20·11-s + (0.693 + 1.20i)13-s + (−0.606 + 1.05i)17-s + (0.917 − 0.397i)19-s + (−0.104 − 0.180i)23-s + (−0.400 − 0.692i)25-s + (0.278 + 0.482i)29-s + 0.718·31-s + 0.328·37-s + (−0.390 + 0.676i)41-s + (0.838 − 1.45i)43-s + (−0.364 − 0.631i)47-s − 49-s + (−0.618 − 1.07i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.125593445\)
\(L(\frac12)\) \(\approx\) \(2.125593445\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.5 + 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-1.5 - 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.367681820947984554783722188979, −8.828028912624420470127963299747, −8.248787913468227015034173976083, −6.80868988005879988007847041521, −6.34505723505851555093738568329, −5.30427868616545111350931758606, −4.44977473391486715702264048887, −3.64121172677040595013894719947, −1.95471061394571443070945452840, −1.16206996123835915376356632410, 1.19434886791347851552481812535, 2.65401172272635213634698606324, 3.32855668736504267473873279740, 4.51418192937962013112816625171, 5.77633603010819234277034443891, 6.29180913169881659201104257972, 7.11649782786393718343444923454, 7.928806002141256890635426187922, 8.991779058125508390947565010689, 9.749983469200895253463882489851

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