Properties

Label 2-1344-56.27-c3-0-86
Degree $2$
Conductor $1344$
Sign $-0.571 + 0.820i$
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 + 15.4·5-s + (6.29 − 17.4i)7-s − 9·9-s − 16.8·11-s + 9.10·13-s − 46.2i·15-s − 73.9i·17-s + 151. i·19-s + (−52.2 − 18.8i)21-s + 0.264i·23-s + 112.·25-s + 27i·27-s − 279. i·29-s + 147.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.37·5-s + (0.339 − 0.940i)7-s − 0.333·9-s − 0.462·11-s + 0.194·13-s − 0.795i·15-s − 1.05i·17-s + 1.82i·19-s + (−0.543 − 0.196i)21-s + 0.00239i·23-s + 0.897·25-s + 0.192i·27-s − 1.79i·29-s + 0.856·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.571 + 0.820i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.571 + 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.419577305\)
\(L(\frac12)\) \(\approx\) \(2.419577305\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.29 + 17.4i)T \)
good5 \( 1 - 15.4T + 125T^{2} \)
11 \( 1 + 16.8T + 1.33e3T^{2} \)
13 \( 1 - 9.10T + 2.19e3T^{2} \)
17 \( 1 + 73.9iT - 4.91e3T^{2} \)
19 \( 1 - 151. iT - 6.85e3T^{2} \)
23 \( 1 - 0.264iT - 1.21e4T^{2} \)
29 \( 1 + 279. iT - 2.43e4T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 + 418. iT - 5.06e4T^{2} \)
41 \( 1 + 164. iT - 6.89e4T^{2} \)
43 \( 1 + 266.T + 7.95e4T^{2} \)
47 \( 1 + 277.T + 1.03e5T^{2} \)
53 \( 1 - 276. iT - 1.48e5T^{2} \)
59 \( 1 - 212. iT - 2.05e5T^{2} \)
61 \( 1 - 174.T + 2.26e5T^{2} \)
67 \( 1 - 317.T + 3.00e5T^{2} \)
71 \( 1 + 106. iT - 3.57e5T^{2} \)
73 \( 1 + 755. iT - 3.89e5T^{2} \)
79 \( 1 + 194. iT - 4.93e5T^{2} \)
83 \( 1 + 997. iT - 5.71e5T^{2} \)
89 \( 1 - 988. iT - 7.04e5T^{2} \)
97 \( 1 + 1.02e3iT - 9.12e5T^{2} \)
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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

−9.007941507733334403977455006928, −7.951382815875139330452511071140, −7.43878740065089690731465005624, −6.35242732168001353732062058237, −5.80357836894868910262419038721, −4.87359212757186483537546982182, −3.70054233536192471683442380076, −2.40839751401485752801952082958, −1.63173103037129866357096781804, −0.51878025386623944697069647341, 1.39604197912654797862707579359, 2.41277280569455083736521201063, 3.21815262748486084644097658134, 4.85006943087924691129695804424, 5.16010860778606633184361668600, 6.16738849931321491973863332055, 6.77705546216509432672875278433, 8.332798115259868366543079718827, 8.695413314690015989321714572542, 9.628080508993485317787850709307

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