Properties

Degree $2$
Conductor $1344$
Sign $-0.826 - 0.563i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 3.15·5-s + (6.11 − 17.4i)7-s − 9·9-s − 60.9·11-s + 59.6·13-s + 9.46i·15-s + 21.4i·17-s − 95.5i·19-s + (52.4 + 18.3i)21-s + 46.3i·23-s − 115.·25-s − 27i·27-s + 107. i·29-s − 94.2·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.282·5-s + (0.329 − 0.944i)7-s − 0.333·9-s − 1.67·11-s + 1.27·13-s + 0.162i·15-s + 0.305i·17-s − 1.15i·19-s + (0.545 + 0.190i)21-s + 0.419i·23-s − 0.920·25-s − 0.192i·27-s + 0.688i·29-s − 0.546·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.826 - 0.563i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.826 - 0.563i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7145302467\)
\(L(\frac12)\) \(\approx\) \(0.7145302467\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3iT \)
7 \( 1 + (-6.11 + 17.4i)T \)
good5 \( 1 - 3.15T + 125T^{2} \)
11 \( 1 + 60.9T + 1.33e3T^{2} \)
13 \( 1 - 59.6T + 2.19e3T^{2} \)
17 \( 1 - 21.4iT - 4.91e3T^{2} \)
19 \( 1 + 95.5iT - 6.85e3T^{2} \)
23 \( 1 - 46.3iT - 1.21e4T^{2} \)
29 \( 1 - 107. iT - 2.43e4T^{2} \)
31 \( 1 + 94.2T + 2.97e4T^{2} \)
37 \( 1 + 131. iT - 5.06e4T^{2} \)
41 \( 1 - 283. iT - 6.89e4T^{2} \)
43 \( 1 - 373.T + 7.95e4T^{2} \)
47 \( 1 + 136.T + 1.03e5T^{2} \)
53 \( 1 - 298. iT - 1.48e5T^{2} \)
59 \( 1 - 468. iT - 2.05e5T^{2} \)
61 \( 1 + 563.T + 2.26e5T^{2} \)
67 \( 1 + 160.T + 3.00e5T^{2} \)
71 \( 1 - 409. iT - 3.57e5T^{2} \)
73 \( 1 - 930. iT - 3.89e5T^{2} \)
79 \( 1 + 442. iT - 4.93e5T^{2} \)
83 \( 1 + 190. iT - 5.71e5T^{2} \)
89 \( 1 - 829. iT - 7.04e5T^{2} \)
97 \( 1 - 1.03e3iT - 9.12e5T^{2} \)
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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

−9.621343580276810259981703025815, −8.800905390494616370093750902197, −7.927904666125376670654278091052, −7.29719134232327530219463671754, −6.11181211652418748715132111197, −5.34463724123891224026319210319, −4.47634339523785329871217376444, −3.58789949351787440965121451963, −2.54471307064216886777821336343, −1.17713449151786176452246168294, 0.16298849599597011194340656693, 1.67851807458978102774278025418, 2.44164935920000651352771471980, 3.51192023966584392973644770421, 4.88525342084439002706489456329, 5.80775671901327796793089516157, 6.12236738443386104220656447130, 7.51301430350633567298130853981, 8.086190804464133954374710809669, 8.717800651453941378341952243309

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