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Properties

Label 2-1368-1368.427-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.755 + 0.654i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (−0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.173 + 0.300i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.939 + 0.342i)18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)6-s + (−0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.173 + 0.300i)11-s + (−0.939 − 0.342i)12-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.939 + 0.342i)18-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (−0.939 + 0.342i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602045141\)
\(L(\frac12)\) \(\approx\) \(1.602045141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.939 + 0.342i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + 1.87T + T^{2} \)
89 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
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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

−9.619456490117970013695261496658, −8.713982535587529758117230094868, −7.68181677585047583123249197623, −6.94098119011121486150418510080, −6.09581860216084658239253413026, −5.30515520588741428781427334483, −4.28947964036128056313822063769, −3.08127421004396755882888516827, −2.34839185203351681768438371412, −1.09625756941108512369313565697, 2.41586123608210884059248430866, 3.48166300700184827308125147625, 4.10053111617777808336367027918, 5.22152524147048888472009234152, 5.62780580823912852260786235550, 6.75592807254274910119958862030, 7.62402841541700319563492272157, 8.564564498685700264987636621585, 9.043788877815124537404811097399, 10.09886937995697438883398568306

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