Properties

Label 2-1344-56.27-c3-0-11
Degree $2$
Conductor $1344$
Sign $-0.631 - 0.775i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + 15.6·5-s + (−10.8 + 15.0i)7-s − 9·9-s − 41.3·11-s + 8.02·13-s − 47.0i·15-s + 14.2i·17-s + 10.2i·19-s + (45.0 + 32.5i)21-s − 203. i·23-s + 121.·25-s + 27i·27-s + 82.6i·29-s + 156.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.40·5-s + (−0.585 + 0.810i)7-s − 0.333·9-s − 1.13·11-s + 0.171·13-s − 0.810i·15-s + 0.202i·17-s + 0.123i·19-s + (0.468 + 0.337i)21-s − 1.84i·23-s + 0.969·25-s + 0.192i·27-s + 0.529i·29-s + 0.905·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.631 - 0.775i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.631 - 0.775i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7250726619\)
\(L(\frac12)\) \(\approx\) \(0.7250726619\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.8 - 15.0i)T \)
good5 \( 1 - 15.6T + 125T^{2} \)
11 \( 1 + 41.3T + 1.33e3T^{2} \)
13 \( 1 - 8.02T + 2.19e3T^{2} \)
17 \( 1 - 14.2iT - 4.91e3T^{2} \)
19 \( 1 - 10.2iT - 6.85e3T^{2} \)
23 \( 1 + 203. iT - 1.21e4T^{2} \)
29 \( 1 - 82.6iT - 2.43e4T^{2} \)
31 \( 1 - 156.T + 2.97e4T^{2} \)
37 \( 1 - 106. iT - 5.06e4T^{2} \)
41 \( 1 - 315. iT - 6.89e4T^{2} \)
43 \( 1 + 154.T + 7.95e4T^{2} \)
47 \( 1 + 474.T + 1.03e5T^{2} \)
53 \( 1 - 266. iT - 1.48e5T^{2} \)
59 \( 1 - 425. iT - 2.05e5T^{2} \)
61 \( 1 + 8.30T + 2.26e5T^{2} \)
67 \( 1 + 820.T + 3.00e5T^{2} \)
71 \( 1 - 109. iT - 3.57e5T^{2} \)
73 \( 1 + 53.8iT - 3.89e5T^{2} \)
79 \( 1 - 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + 507. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 + 573. iT - 9.12e5T^{2} \)
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

−9.620184974304895416674072180530, −8.636973174311779024368240725447, −8.118809616543252204238885326193, −6.78689794591056815718675273680, −6.24640896138433057790971180257, −5.57283646577125908070005613638, −4.71755886842369945271122425326, −2.92540609942541689406859855895, −2.47361702723809010845768584148, −1.37202680962342255365203954661, 0.14991983195575742606897298036, 1.62133613807346604380769148486, 2.76153947964473270581030131331, 3.65095284703160781629904822336, 4.86921114183428745928145428840, 5.57057748746664058737242645942, 6.32054400885132528513477052046, 7.28673929233823978939577911009, 8.175735598033670154699954266337, 9.331452833751330975318611418097

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