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Properties

Label 2-1368-1368.691-c0-0-2
Degree $2$
Conductor $1368$
Sign $-0.790 + 0.612i$
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  − i·2-s − 3-s − 4-s + (−0.866 − 0.5i)5-s + i·6-s + (0.866 + 0.5i)7-s + i·8-s + 9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 12-s + (0.5 − 0.866i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s − i·18-s + ⋯
L(s)  = 1  − i·2-s − 3-s − 4-s + (−0.866 − 0.5i)5-s + i·6-s + (0.866 + 0.5i)7-s + i·8-s + 9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 12-s + (0.5 − 0.866i)14-s + (0.866 + 0.5i)15-s + 16-s + (−0.5 − 0.866i)17-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5927922288\)
\(L(\frac12)\) \(\approx\) \(0.5927922288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + T \)
19 \( 1 - T \)
good5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.353326675013679398010100654638, −8.969078739805931645637729353509, −7.950783588858228305544409918211, −7.18651228174847546650366281945, −5.68533425837039125906128231297, −5.20759666615194740125820140503, −4.30356943788939694742619366100, −3.50908298697910065660798941257, −1.93328793093652259146125002360, −0.64883884252614936288081504103, 1.41040769341772955298021833895, 3.70736182725343264273357302482, 4.37402831368063362271674168739, 5.08414426631715580775427205765, 6.14046629083461239473195480549, 6.88338426545065939874180397311, 7.61761555947485876040635225999, 7.996315755544614159341528969600, 9.357892879685720733896280210297, 9.997845002965037758416562084212

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