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Properties

Label 2-1344-56.27-c3-0-53
Degree $2$
Conductor $1344$
Sign $-0.249 - 0.968i$
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 + 10.4·5-s + (−9.40 + 15.9i)7-s − 9·9-s + 51.5·11-s + 46.4·13-s + 31.2i·15-s + 99.1i·17-s + 12.5i·19-s + (−47.8 − 28.2i)21-s + 129. i·23-s − 16.2·25-s − 27i·27-s − 207. i·29-s + 139.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.932·5-s + (−0.507 + 0.861i)7-s − 0.333·9-s + 1.41·11-s + 0.990·13-s + 0.538i·15-s + 1.41i·17-s + 0.151i·19-s + (−0.497 − 0.293i)21-s + 1.17i·23-s − 0.130·25-s − 0.192i·27-s − 1.32i·29-s + 0.808·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.249 - 0.968i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.249 - 0.968i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.730311975\)
\(L(\frac12)\) \(\approx\) \(2.730311975\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.40 - 15.9i)T \)
good5 \( 1 - 10.4T + 125T^{2} \)
11 \( 1 - 51.5T + 1.33e3T^{2} \)
13 \( 1 - 46.4T + 2.19e3T^{2} \)
17 \( 1 - 99.1iT - 4.91e3T^{2} \)
19 \( 1 - 12.5iT - 6.85e3T^{2} \)
23 \( 1 - 129. iT - 1.21e4T^{2} \)
29 \( 1 + 207. iT - 2.43e4T^{2} \)
31 \( 1 - 139.T + 2.97e4T^{2} \)
37 \( 1 - 152. iT - 5.06e4T^{2} \)
41 \( 1 + 317. iT - 6.89e4T^{2} \)
43 \( 1 - 366.T + 7.95e4T^{2} \)
47 \( 1 + 103.T + 1.03e5T^{2} \)
53 \( 1 - 24.2iT - 1.48e5T^{2} \)
59 \( 1 - 339. iT - 2.05e5T^{2} \)
61 \( 1 - 25.3T + 2.26e5T^{2} \)
67 \( 1 - 660.T + 3.00e5T^{2} \)
71 \( 1 - 0.771iT - 3.57e5T^{2} \)
73 \( 1 - 342. iT - 3.89e5T^{2} \)
79 \( 1 - 951. iT - 4.93e5T^{2} \)
83 \( 1 + 244. iT - 5.71e5T^{2} \)
89 \( 1 + 1.17e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.40e3iT - 9.12e5T^{2} \)
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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

−9.541557916748759314195200908212, −8.840855708957576567696362605990, −8.162990250000124788741723274803, −6.68836085361577726479743085804, −5.95848510205304397591403158673, −5.67218188340948207315594393132, −4.17011272348047392869660026693, −3.54299594881301012866179174989, −2.26012092048856258370162775697, −1.27456675811989961902251638493, 0.67625249574060013954071434579, 1.44252528580388345078188834078, 2.70422008641707181149348239030, 3.73379137802833511683623876817, 4.76834964277394450800617267269, 5.98093544189519283044371693846, 6.58097660749050111684422220593, 7.09341384761854638678044396718, 8.229853842824165987272367810044, 9.196234927659906205183265364180

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