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Properties

Label 2-1368-1368.851-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.919 - 0.392i$
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  + (0.173 + 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (1.11 + 0.642i)11-s + (−0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)18-s + (0.939 − 0.342i)19-s + (−0.439 + 1.20i)22-s − 0.999·24-s + (0.939 − 0.342i)25-s − 0.999·27-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.5 − 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)6-s + (−0.5 − 0.866i)8-s + (−0.499 − 0.866i)9-s + (1.11 + 0.642i)11-s + (−0.173 + 0.984i)12-s + (0.766 − 0.642i)16-s + (0.766 − 0.642i)18-s + (0.939 − 0.342i)19-s + (−0.439 + 1.20i)22-s − 0.999·24-s + (0.939 − 0.342i)25-s − 0.999·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.919 - 0.392i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.919 - 0.392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311483259\)
\(L(\frac12)\) \(\approx\) \(1.311483259\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
good5 \( 1 + (-0.939 + 0.342i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
61 \( 1 + (0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.173 - 0.984i)T^{2} \)
83 \( 1 - 1.28iT - T^{2} \)
89 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.673 + 0.118i)T + (0.939 - 0.342i)T^{2} \)
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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

−9.237201502688920725393816334257, −9.075386777039724975616636282739, −8.020306493462425159742479097817, −7.30331333765544819726464347704, −6.72242546277859884890223072897, −5.98691541495098105323973191745, −4.88842131975179423358467674332, −3.89074995902673990414374412984, −2.86862437622946426643500445896, −1.27529005991150090955010843577, 1.45669387877721649698146282920, 2.88518491085101072266911213239, 3.53708868994090588011597087913, 4.39167303576324720357072464115, 5.26190890977076436086247602936, 6.16050651628753619347681358638, 7.54587833097219828014355349461, 8.612955712022990144164087233409, 9.043817182525878276096652376431, 9.786122968663450037612546124645

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