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Properties

Label 2-1368-1368.1147-c0-0-1
Degree $2$
Conductor $1368$
Sign $0.532 - 0.846i$
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  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + 0.999·12-s − 0.999·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + i·5-s + (0.866 − 0.499i)6-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + 0.999·12-s − 0.999·14-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.532 - 0.846i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.532 - 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.906287018\)
\(L(\frac12)\) \(\approx\) \(1.906287018\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 - T \)
good5 \( 1 - iT - T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.645918546414351168286465201181, −9.100060567986719593763359230471, −7.905107337216527733307352372442, −7.25734553402050569220586194631, −6.61305250744214630442010738315, −6.16080649644217841134649876789, −4.95740089069718802334996649213, −3.65098847549619105419382397698, −2.91939561566138770044117871976, −2.16692757118555635312731075418, 1.30411642960681408741818058422, 2.93934470260157838031725274516, 3.60754821405208334200318113799, 4.42289766872223933828040861818, 5.20572034726390215150643826271, 6.08633487566498625649392253339, 7.01914441030704218190678910829, 8.414123890287504828742761318302, 8.913078008207208663864727368312, 9.953801215529687821272760207540

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