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Properties

Label 2-1368-1368.1147-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9453931494\)
\(L(\frac12)\) \(\approx\) \(0.9453931494\)
\(L(1)\) 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

−9.276304607043640444180265938936, −8.772470473703214273557774799739, −8.018605154877109010856550413784, −7.27709632304256736128700877789, −6.76236065807928440889998153617, −5.20243710816303166687612278912, −4.26188391948665982118974664611, −3.06993980804925259357344853093, −1.79110014118255128023250187046, −1.11509134424037608747398932288, 1.87475071960763901668629763110, 2.91143818725057164808957848784, 4.01531785015958833579878524333, 5.30811461816349690674588801138, 5.91853981482972686789286921762, 6.99216523497854395281436382116, 7.82137499720435240944251594843, 8.593366335064811441943201602690, 9.071088563308558825389833910668, 9.996492494032903313097783559165

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