Properties

Degree $2$
Conductor $1344$
Sign $-0.996 - 0.0868i$
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 + 12.8·5-s + (−6.32 + 17.4i)7-s − 9·9-s + 25.0·11-s − 64.1·13-s − 38.4i·15-s + 77.4i·17-s − 49.9i·19-s + (52.2 + 18.9i)21-s + 80.5i·23-s + 39.4·25-s + 27i·27-s − 159. i·29-s − 96.5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.14·5-s + (−0.341 + 0.939i)7-s − 0.333·9-s + 0.686·11-s − 1.36·13-s − 0.662i·15-s + 1.10i·17-s − 0.603i·19-s + (0.542 + 0.197i)21-s + 0.730i·23-s + 0.315·25-s + 0.192i·27-s − 1.01i·29-s − 0.559·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.996 - 0.0868i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.996 - 0.0868i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.02135183617\)
\(L(\frac12)\) \(\approx\) \(0.02135183617\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.32 - 17.4i)T \)
good5 \( 1 - 12.8T + 125T^{2} \)
11 \( 1 - 25.0T + 1.33e3T^{2} \)
13 \( 1 + 64.1T + 2.19e3T^{2} \)
17 \( 1 - 77.4iT - 4.91e3T^{2} \)
19 \( 1 + 49.9iT - 6.85e3T^{2} \)
23 \( 1 - 80.5iT - 1.21e4T^{2} \)
29 \( 1 + 159. iT - 2.43e4T^{2} \)
31 \( 1 + 96.5T + 2.97e4T^{2} \)
37 \( 1 + 274. iT - 5.06e4T^{2} \)
41 \( 1 - 299. iT - 6.89e4T^{2} \)
43 \( 1 + 385.T + 7.95e4T^{2} \)
47 \( 1 + 418.T + 1.03e5T^{2} \)
53 \( 1 + 665. iT - 1.48e5T^{2} \)
59 \( 1 - 445. iT - 2.05e5T^{2} \)
61 \( 1 + 599.T + 2.26e5T^{2} \)
67 \( 1 - 675.T + 3.00e5T^{2} \)
71 \( 1 - 877. iT - 3.57e5T^{2} \)
73 \( 1 + 696. iT - 3.89e5T^{2} \)
79 \( 1 + 1.24e3iT - 4.93e5T^{2} \)
83 \( 1 + 238. iT - 5.71e5T^{2} \)
89 \( 1 + 743. iT - 7.04e5T^{2} \)
97 \( 1 + 1.55e3iT - 9.12e5T^{2} \)
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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

−8.954642170435430727233616790887, −8.052466314581413952246932394961, −7.04160912103423848558096862763, −6.25165868067078201441845121849, −5.69981393870740356878420204052, −4.78905506510578620034752830602, −3.33742629215002672841190178939, −2.23983633656523297153183146297, −1.68808869899923630828908458598, −0.00433674185245126589541753958, 1.41584882816721148072420387846, 2.61368447710084743882217206902, 3.60870053917279667190656826836, 4.71732578905697948254566673507, 5.31858957727880658245944243424, 6.48668328251658784990572401403, 7.00565019680415338179315458564, 8.064550648216045503138368764495, 9.262305744474746621782156647198, 9.643564488300076437327716762116

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