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Properties

Label 2-1344-28.27-c3-0-92
Degree $2$
Conductor $1344$
Sign $-0.953 + 0.300i$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 2.26i·5-s + (−5.56 − 17.6i)7-s + 9·9-s − 56.3i·11-s − 19.3i·13-s − 6.79i·15-s − 127. i·17-s − 5.09·19-s + (−16.6 − 52.9i)21-s + 94.6i·23-s + 119.·25-s + 27·27-s + 92.4·29-s − 38.9·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.202i·5-s + (−0.300 − 0.953i)7-s + 0.333·9-s − 1.54i·11-s − 0.413i·13-s − 0.116i·15-s − 1.81i·17-s − 0.0615·19-s + (−0.173 − 0.550i)21-s + 0.857i·23-s + 0.958·25-s + 0.192·27-s + 0.592·29-s − 0.225·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.953 + 0.300i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.953 + 0.300i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.687089636\)
\(L(\frac12)\) \(\approx\) \(1.687089636\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 + (5.56 + 17.6i)T \)
good5 \( 1 + 2.26iT - 125T^{2} \)
11 \( 1 + 56.3iT - 1.33e3T^{2} \)
13 \( 1 + 19.3iT - 2.19e3T^{2} \)
17 \( 1 + 127. iT - 4.91e3T^{2} \)
19 \( 1 + 5.09T + 6.85e3T^{2} \)
23 \( 1 - 94.6iT - 1.21e4T^{2} \)
29 \( 1 - 92.4T + 2.43e4T^{2} \)
31 \( 1 + 38.9T + 2.97e4T^{2} \)
37 \( 1 + 290.T + 5.06e4T^{2} \)
41 \( 1 - 118. iT - 6.89e4T^{2} \)
43 \( 1 - 274. iT - 7.95e4T^{2} \)
47 \( 1 + 612.T + 1.03e5T^{2} \)
53 \( 1 + 55.9T + 1.48e5T^{2} \)
59 \( 1 - 701.T + 2.05e5T^{2} \)
61 \( 1 - 655. iT - 2.26e5T^{2} \)
67 \( 1 - 109. iT - 3.00e5T^{2} \)
71 \( 1 + 478. iT - 3.57e5T^{2} \)
73 \( 1 + 562. iT - 3.89e5T^{2} \)
79 \( 1 + 312. iT - 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 + 743. iT - 7.04e5T^{2} \)
97 \( 1 + 1.27e3iT - 9.12e5T^{2} \)
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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

−8.844614326862637988996119767872, −8.124061771374253512722556747066, −7.28935274113449629835198739992, −6.57757593689308543638245856590, −5.43948757715701314209881007442, −4.58211325767572361434808397857, −3.34913594000364419052613553419, −2.96812162958408805926943101439, −1.24937655535223498358927572999, −0.35486502288973021809525872021, 1.69581130853802024076870177708, 2.35375267479263166736381903304, 3.50133464321023604211219770479, 4.44866139940871302292438107087, 5.38485979659005285879009681031, 6.58672989921568163933439076415, 6.97197449098167044869135707719, 8.311865547181749498798041777980, 8.610083381155312774780387253047, 9.651169073519314550536570311931

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