Properties

Label 2-1368-1368.1075-c0-0-1
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: $\chi_{1368} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7663312081\)
\(L(\frac12)\) \(\approx\) \(0.7663312081\)
\(L(1)\) 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

−9.706524933102207213809161185769, −9.406618111709471746522543570308, −8.420597764420187083454370283684, −7.05817390044080873027146028106, −6.16295610399732562447420121596, −5.43551125392601401274057227010, −4.66874248408021146650973203897, −3.65820971692902584889112492813, −2.28048000810519457532227374482, −1.25703114398186467022443669450, 0.891104969671182459887282015517, 2.99455399907437500758744137668, 4.15116533850862566982755605435, 5.12940009761133881595293212754, 6.00881636040310916099743565699, 6.51697632667045725581370962330, 7.00975701338042060212418530383, 8.076604005165994214533791791412, 9.238362244819741113297969824314, 9.888207920314216324514564461299

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