Properties

Degree $2$
Conductor $1344$
Sign $0.355 - 0.934i$
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 − 15.6·5-s + (−10.8 + 15.0i)7-s − 9·9-s + 41.3·11-s − 8.02·13-s − 47.0i·15-s + 14.2i·17-s − 10.2i·19-s + (−45.0 − 32.5i)21-s − 203. i·23-s + 121.·25-s − 27i·27-s − 82.6i·29-s + 156.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.40·5-s + (−0.585 + 0.810i)7-s − 0.333·9-s + 1.13·11-s − 0.171·13-s − 0.810i·15-s + 0.202i·17-s − 0.123i·19-s + (−0.468 − 0.337i)21-s − 1.84i·23-s + 0.969·25-s − 0.192i·27-s − 0.529i·29-s + 0.905·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.355 - 0.934i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.355 - 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.157607799\)
\(L(\frac12)\) \(\approx\) \(1.157607799\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.8 - 15.0i)T \)
good5 \( 1 + 15.6T + 125T^{2} \)
11 \( 1 - 41.3T + 1.33e3T^{2} \)
13 \( 1 + 8.02T + 2.19e3T^{2} \)
17 \( 1 - 14.2iT - 4.91e3T^{2} \)
19 \( 1 + 10.2iT - 6.85e3T^{2} \)
23 \( 1 + 203. iT - 1.21e4T^{2} \)
29 \( 1 + 82.6iT - 2.43e4T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 + 106. iT - 5.06e4T^{2} \)
41 \( 1 - 315. iT - 6.89e4T^{2} \)
43 \( 1 - 154.T + 7.95e4T^{2} \)
47 \( 1 + 474.T + 1.03e5T^{2} \)
53 \( 1 + 266. iT - 1.48e5T^{2} \)
59 \( 1 + 425. iT - 2.05e5T^{2} \)
61 \( 1 - 8.30T + 2.26e5T^{2} \)
67 \( 1 - 820.T + 3.00e5T^{2} \)
71 \( 1 - 109. iT - 3.57e5T^{2} \)
73 \( 1 + 53.8iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 - 507. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 + 573. iT - 9.12e5T^{2} \)
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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

−9.396906051153358205789917720490, −8.487922759336698254961530840633, −8.087463313586238786256461597916, −6.76838910941503057042527771014, −6.27937776109648199341620854397, −4.96844233683508272087355374123, −4.18596802307453414257640519605, −3.46678869024292099437041135142, −2.45319352698250395151211133912, −0.63896254204605106055788561395, 0.47315616275243775188083278770, 1.47394348748613471969318116349, 3.18357127920071955968996670156, 3.76445986648910293752910474100, 4.63868312488657841298738636871, 5.95324291738497182225894414766, 6.92528699061425457591362039757, 7.35353858521797006307491866877, 8.085822262895768401304840072253, 9.020101418035666833857887993827

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