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Properties

Label 2-1368-57.8-c1-0-12
Degree $2$
Conductor $1368$
Sign $0.932 - 0.361i$
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  + (1.76 + 1.01i)5-s + 3.69·7-s + 0.605i·11-s + (2.65 − 1.53i)13-s + (2.40 + 1.39i)17-s + (0.780 + 4.28i)19-s + (−2.23 + 1.29i)23-s + (−0.422 − 0.732i)25-s + (−1.70 − 2.95i)29-s − 4.42i·31-s + (6.52 + 3.76i)35-s − 3.34i·37-s + (−1.50 + 2.60i)41-s + (0.562 − 0.974i)43-s + (−3.05 + 1.76i)47-s + ⋯
L(s)  = 1  + (0.789 + 0.455i)5-s + 1.39·7-s + 0.182i·11-s + (0.735 − 0.424i)13-s + (0.584 + 0.337i)17-s + (0.179 + 0.983i)19-s + (−0.466 + 0.269i)23-s + (−0.0845 − 0.146i)25-s + (−0.316 − 0.548i)29-s − 0.795i·31-s + (1.10 + 0.637i)35-s − 0.550i·37-s + (−0.234 + 0.406i)41-s + (0.0858 − 0.148i)43-s + (−0.445 + 0.257i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.932 - 0.361i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.932 - 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.365583475\)
\(L(\frac12)\) \(\approx\) \(2.365583475\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.780 - 4.28i)T \)
good5 \( 1 + (-1.76 - 1.01i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.69T + 7T^{2} \)
11 \( 1 - 0.605iT - 11T^{2} \)
13 \( 1 + (-2.65 + 1.53i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.40 - 1.39i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.23 - 1.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.70 + 2.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.42iT - 31T^{2} \)
37 \( 1 + 3.34iT - 37T^{2} \)
41 \( 1 + (1.50 - 2.60i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.562 + 0.974i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.05 - 1.76i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.984 + 1.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.48 - 6.03i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.98 - 8.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.324 + 0.187i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.03 - 13.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.09 + 14.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.44 + 4.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.17iT - 83T^{2} \)
89 \( 1 + (3.91 + 6.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.57 - 4.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.879348382572559811451261692135, −8.712299769867306274459550560349, −7.998644056660772623845940558759, −7.39087086698156808465408692404, −6.01282297248871911593617712922, −5.73849293950491643639500288104, −4.56853752946523928658588360398, −3.58616830110071391314633587596, −2.24173071224062025809909071788, −1.38843832938046539710266750115, 1.20085547884877524232856404213, 2.04639951174449388953316414251, 3.44509024839763942295604898356, 4.72804142764161709300801363780, 5.20765137735101562789813727620, 6.14337762770150001209124349440, 7.13621211037694498063841000923, 8.078889079177828949796400467360, 8.747589652325711444278488779304, 9.415443402458369622184849771898

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