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Properties

Label 2-1344-56.27-c3-0-19
Degree $2$
Conductor $1344$
Sign $-0.851 - 0.523i$
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  + 3i·3-s − 10.6·5-s + (−18.0 + 4.29i)7-s − 9·9-s + 5.02·11-s + 80.6·13-s − 31.9i·15-s + 31.3i·17-s − 74.0i·19-s + (−12.8 − 54.0i)21-s + 37.3i·23-s − 11.9·25-s − 27i·27-s + 242. i·29-s + 284.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.951·5-s + (−0.972 + 0.231i)7-s − 0.333·9-s + 0.137·11-s + 1.72·13-s − 0.549i·15-s + 0.446i·17-s − 0.894i·19-s + (−0.133 − 0.561i)21-s + 0.338i·23-s − 0.0952·25-s − 0.192i·27-s + 1.55i·29-s + 1.65·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.851 - 0.523i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.851 - 0.523i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8909346357\)
\(L(\frac12)\) \(\approx\) \(0.8909346357\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.0 - 4.29i)T \)
good5 \( 1 + 10.6T + 125T^{2} \)
11 \( 1 - 5.02T + 1.33e3T^{2} \)
13 \( 1 - 80.6T + 2.19e3T^{2} \)
17 \( 1 - 31.3iT - 4.91e3T^{2} \)
19 \( 1 + 74.0iT - 6.85e3T^{2} \)
23 \( 1 - 37.3iT - 1.21e4T^{2} \)
29 \( 1 - 242. iT - 2.43e4T^{2} \)
31 \( 1 - 284.T + 2.97e4T^{2} \)
37 \( 1 + 167. iT - 5.06e4T^{2} \)
41 \( 1 + 270. iT - 6.89e4T^{2} \)
43 \( 1 + 169.T + 7.95e4T^{2} \)
47 \( 1 - 639.T + 1.03e5T^{2} \)
53 \( 1 - 534. iT - 1.48e5T^{2} \)
59 \( 1 + 56.9iT - 2.05e5T^{2} \)
61 \( 1 + 217.T + 2.26e5T^{2} \)
67 \( 1 + 684.T + 3.00e5T^{2} \)
71 \( 1 + 966. iT - 3.57e5T^{2} \)
73 \( 1 - 325. iT - 3.89e5T^{2} \)
79 \( 1 - 856. iT - 4.93e5T^{2} \)
83 \( 1 - 459. iT - 5.71e5T^{2} \)
89 \( 1 - 1.07e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.22e3iT - 9.12e5T^{2} \)
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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

−9.358158789534401339729205498244, −8.876574487734496014571618752044, −8.127932476858706822554107628453, −7.06845816970958300564011542999, −6.27424462583618755478587865044, −5.44278871845665654184854027265, −4.17304334346931230814344710373, −3.66663242462825805400345004080, −2.77810166536469584131792588610, −1.02495071315818445591515757467, 0.26303482514029348748619120498, 1.24587402883796508207114591150, 2.79012789424381942509403925235, 3.67478473159232220584877076934, 4.38260582943102421520748617580, 5.97002600327895592409252868808, 6.34057792976174541705751020157, 7.31764745116949115642475555245, 8.130831979961643319388648194150, 8.643914602234395368643714782736

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