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Properties

Label 2-1368-1368.283-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.776 + 0.630i$
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  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s − 1.87·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.347 − 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s − 1.87·11-s + (−0.939 + 0.342i)12-s + (0.173 − 0.984i)16-s + (0.347 − 1.96i)17-s + (−0.939 − 0.342i)18-s + (0.173 + 0.984i)19-s + (1.76 − 0.642i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.776 + 0.630i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.776 + 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1622464514\)
\(L(\frac12)\) \(\approx\) \(0.1622464514\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 1.87T + T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.347 + 1.96i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.778811967377034943473399621945, −8.473349002577420088396669849758, −7.66869407674393974455403807767, −7.28968139882676662706049221602, −6.24065540826120027049745277935, −5.39521253728323000543857368702, −4.91672145825870559323984662226, −2.97927172552445241011113117541, −1.83582363627582254173468044454, −0.19785671053864246233162746748, 1.62327040332463009654822383384, 2.95612572687596962168003483353, 4.05474161767340251238105617188, 5.24428967078670367066662669490, 6.00531121013567481306194035779, 6.94656132317318540717616304132, 7.83051447252347100912182512786, 8.458515105810944955369685820120, 9.551844332534569212555842754037, 10.28121779564040163535543332555

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