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Properties

Label 2-1344-28.27-c3-0-25
Degree $2$
Conductor $1344$
Sign $-0.955 + 0.295i$
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 + 20.9i·5-s + (−17.6 + 5.46i)7-s + 9·9-s + 23.8i·11-s + 74.6i·13-s + 62.7i·15-s + 68.6i·17-s + 26.6·19-s + (−53.0 + 16.4i)21-s + 74.6i·23-s − 313.·25-s + 27·27-s − 128.·29-s + 212.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.87i·5-s + (−0.955 + 0.295i)7-s + 0.333·9-s + 0.653i·11-s + 1.59i·13-s + 1.08i·15-s + 0.979i·17-s + 0.321·19-s + (−0.551 + 0.170i)21-s + 0.677i·23-s − 2.50·25-s + 0.192·27-s − 0.822·29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.955 + 0.295i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.955 + 0.295i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.887536942\)
\(L(\frac12)\) \(\approx\) \(1.887536942\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.6 - 5.46i)T \)
good5 \( 1 - 20.9iT - 125T^{2} \)
11 \( 1 - 23.8iT - 1.33e3T^{2} \)
13 \( 1 - 74.6iT - 2.19e3T^{2} \)
17 \( 1 - 68.6iT - 4.91e3T^{2} \)
19 \( 1 - 26.6T + 6.85e3T^{2} \)
23 \( 1 - 74.6iT - 1.21e4T^{2} \)
29 \( 1 + 128.T + 2.43e4T^{2} \)
31 \( 1 - 212.T + 2.97e4T^{2} \)
37 \( 1 - 329.T + 5.06e4T^{2} \)
41 \( 1 - 182. iT - 6.89e4T^{2} \)
43 \( 1 + 260. iT - 7.95e4T^{2} \)
47 \( 1 + 401.T + 1.03e5T^{2} \)
53 \( 1 - 76.7T + 1.48e5T^{2} \)
59 \( 1 - 901.T + 2.05e5T^{2} \)
61 \( 1 - 271. iT - 2.26e5T^{2} \)
67 \( 1 + 499. iT - 3.00e5T^{2} \)
71 \( 1 - 299. iT - 3.57e5T^{2} \)
73 \( 1 - 452. iT - 3.89e5T^{2} \)
79 \( 1 + 347. iT - 4.93e5T^{2} \)
83 \( 1 + 775.T + 5.71e5T^{2} \)
89 \( 1 - 48.7iT - 7.04e5T^{2} \)
97 \( 1 + 1.05e3iT - 9.12e5T^{2} \)
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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

−9.910080503260126787609078239335, −9.041537500236008240228073628168, −7.918550286659661257083774366607, −7.04829614039483841774533144979, −6.63386859601521133813684732153, −5.86930926547000160028087207574, −4.22461617565093211825059791061, −3.52241039912247784893299091864, −2.63434942240192698607773271664, −1.86286012341735982877662995501, 0.46475500167690155037525686253, 0.958737551360815106508941007378, 2.61415572333668076773625528578, 3.53031388713522132139007640467, 4.54249245630721320632981515085, 5.37045353971847655822189487839, 6.15353708874863028879231739961, 7.43079758449927264419761831167, 8.201285246522442518034722385606, 8.696816625732099602494981554773

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