Properties

Label 2-1344-28.27-c3-0-60
Degree $2$
Conductor $1344$
Sign $0.368 + 0.929i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 15.0i·5-s + (−17.2 + 6.82i)7-s + 9·9-s + 27.2i·11-s + 25.6i·13-s − 45.1i·15-s − 13.7i·17-s + 147.·19-s + (−51.6 + 20.4i)21-s − 87.4i·23-s − 101.·25-s + 27·27-s + 19.9·29-s + 24.8·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34i·5-s + (−0.929 + 0.368i)7-s + 0.333·9-s + 0.745i·11-s + 0.546i·13-s − 0.777i·15-s − 0.196i·17-s + 1.77·19-s + (−0.536 + 0.212i)21-s − 0.792i·23-s − 0.812·25-s + 0.192·27-s + 0.127·29-s + 0.143·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.368 + 0.929i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.368 + 0.929i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.256573498\)
\(L(\frac12)\) \(\approx\) \(2.256573498\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.2 - 6.82i)T \)
good5 \( 1 + 15.0iT - 125T^{2} \)
11 \( 1 - 27.2iT - 1.33e3T^{2} \)
13 \( 1 - 25.6iT - 2.19e3T^{2} \)
17 \( 1 + 13.7iT - 4.91e3T^{2} \)
19 \( 1 - 147.T + 6.85e3T^{2} \)
23 \( 1 + 87.4iT - 1.21e4T^{2} \)
29 \( 1 - 19.9T + 2.43e4T^{2} \)
31 \( 1 - 24.8T + 2.97e4T^{2} \)
37 \( 1 + 338.T + 5.06e4T^{2} \)
41 \( 1 - 195. iT - 6.89e4T^{2} \)
43 \( 1 + 216. iT - 7.95e4T^{2} \)
47 \( 1 - 527.T + 1.03e5T^{2} \)
53 \( 1 - 371.T + 1.48e5T^{2} \)
59 \( 1 + 77.9T + 2.05e5T^{2} \)
61 \( 1 + 671. iT - 2.26e5T^{2} \)
67 \( 1 - 706. iT - 3.00e5T^{2} \)
71 \( 1 + 1.06e3iT - 3.57e5T^{2} \)
73 \( 1 - 254. iT - 3.89e5T^{2} \)
79 \( 1 - 38.8iT - 4.93e5T^{2} \)
83 \( 1 + 33.1T + 5.71e5T^{2} \)
89 \( 1 + 188. iT - 7.04e5T^{2} \)
97 \( 1 + 1.34e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.079619821648511381102386651309, −8.540856233043013301114311670551, −7.48101891110173855045227673490, −6.77284236800461332082335231444, −5.58551240203725202684108895592, −4.85155661580762094448257979795, −3.92688500009959109580205101814, −2.87509941099186510148242725537, −1.73498209090107428651144428862, −0.57914957150779770040169379661, 0.955023807279980225806983643600, 2.57208428308574458765040961330, 3.28690017225958774357190133618, 3.75333771931073815985815253358, 5.40038280334925159744628452121, 6.18656372292944313663411226070, 7.20273225074807601557743121973, 7.45922500074986413501021693907, 8.634994819784321836134248755302, 9.489278800860534012855189289833

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