↑

Properties

Label 2-1344-28.27-c3-0-44
Degree $2$
Conductor $1344$
Sign $0.820 - 0.571i$
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 − 4.15i·5-s + (10.5 + 15.1i)7-s + 9·9-s + 26.7i·11-s − 36.8i·13-s − 12.4i·15-s − 41.0i·17-s − 110.·19-s + (31.7 + 45.5i)21-s + 69.5i·23-s + 107.·25-s + 27·27-s + 155.·29-s + 98.2·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.371i·5-s + (0.571 + 0.820i)7-s + 0.333·9-s + 0.732i·11-s − 0.786i·13-s − 0.214i·15-s − 0.585i·17-s − 1.33·19-s + (0.330 + 0.473i)21-s + 0.630i·23-s + 0.861·25-s + 0.192·27-s + 0.992·29-s + 0.569·31-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(2.870019818\)
\(L(\frac12)\) \(\approx\) \(2.870019818\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.5 - 15.1i)T \)
good5 \( 1 + 4.15iT - 125T^{2} \)
11 \( 1 - 26.7iT - 1.33e3T^{2} \)
13 \( 1 + 36.8iT - 2.19e3T^{2} \)
17 \( 1 + 41.0iT - 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
23 \( 1 - 69.5iT - 1.21e4T^{2} \)
29 \( 1 - 155.T + 2.43e4T^{2} \)
31 \( 1 - 98.2T + 2.97e4T^{2} \)
37 \( 1 - 210.T + 5.06e4T^{2} \)
41 \( 1 - 0.600iT - 6.89e4T^{2} \)
43 \( 1 - 354. iT - 7.95e4T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 - 274.T + 1.48e5T^{2} \)
59 \( 1 + 301.T + 2.05e5T^{2} \)
61 \( 1 + 469. iT - 2.26e5T^{2} \)
67 \( 1 + 605. iT - 3.00e5T^{2} \)
71 \( 1 - 497. iT - 3.57e5T^{2} \)
73 \( 1 - 429. iT - 3.89e5T^{2} \)
79 \( 1 - 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 936. iT - 7.04e5T^{2} \)
97 \( 1 - 407. iT - 9.12e5T^{2} \)
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

−9.235827018351950626565615392101, −8.437377522661109700257084492798, −7.963725574190740125959990305528, −6.95958754459563156412412061419, −5.96743344328213115790979836821, −4.95111479890404708361478144100, −4.36030298623147188362579442917, −2.96053685981959665076906652349, −2.21264313204873466691707890163, −1.01044211846203983162279819134, 0.71157088761413102676407321789, 1.93076531588055702801081479830, 2.96012310136144002900191212069, 4.08912410524889256543729631023, 4.59961758544777243650319664021, 6.04771331690807481374593001532, 6.77203490423668790240636313939, 7.55775017773446314245488582258, 8.571811188006841682384738411190, 8.796059477486307051949658069559

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