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Properties

Label 2-1368-1368.1307-c0-0-1
Degree $2$
Conductor $1368$
Sign $-0.452 + 0.891i$
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 − 3-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + 9-s − 0.684i·11-s + (−0.173 + 0.984i)12-s + (−0.939 − 0.342i)16-s + (0.592 − 1.62i)17-s + (0.766 − 0.642i)18-s + (−0.173 + 0.984i)19-s + (−0.439 − 0.524i)22-s + (0.500 + 0.866i)24-s + (−0.766 − 0.642i)25-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s − 3-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)6-s + (−0.500 − 0.866i)8-s + 9-s − 0.684i·11-s + (−0.173 + 0.984i)12-s + (−0.939 − 0.342i)16-s + (0.592 − 1.62i)17-s + (0.766 − 0.642i)18-s + (−0.173 + 0.984i)19-s + (−0.439 − 0.524i)22-s + (0.500 + 0.866i)24-s + (−0.766 − 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.123642302\)
\(L(\frac12)\) \(\approx\) \(1.123642302\)
\(L(1)\) 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 \( 1 + T \)
19 \( 1 + (0.173 - 0.984i)T \)
good5 \( 1 + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + 0.684iT - T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (0.939 + 0.342i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.347 + 1.96i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
61 \( 1 + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (0.439 - 0.524i)T + (-0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
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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

−9.872753009095947601101015777709, −9.019025428462849275155805269778, −7.65893978601297713622308396450, −6.83849645864095500132667735449, −5.81689593578337409144251652239, −5.47011449175016314668827736227, −4.40770287557479677120210061926, −3.57964509186964109195549706119, −2.30881956085845287863328944923, −0.862286479228001711428707104195, 1.84054963731599005273577911957, 3.39449646324545225796979668459, 4.40035414254012509797823602235, 5.01612820964323366732857670994, 6.05419141831477764790828635024, 6.45582348949527323806379725548, 7.48928603042435754971644403814, 8.050855926725205333715794144918, 9.276865428825989572970320387813, 10.10076185682698760730284311655

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