Properties

Label 2-1344-8.5-c3-0-8
Degree $2$
Conductor $1344$
Sign $-0.707 + 0.707i$
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  + 3i·3-s + 15.6i·5-s + 7·7-s − 9·9-s + 13.3i·11-s + 7.74i·13-s − 46.8·15-s − 25.3·17-s + 71.9i·19-s + 21i·21-s + 23.6·23-s − 118.·25-s − 27i·27-s − 9.49i·29-s − 74.7·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.39i·5-s + 0.377·7-s − 0.333·9-s + 0.366i·11-s + 0.165i·13-s − 0.806·15-s − 0.361·17-s + 0.869i·19-s + 0.218i·21-s + 0.214·23-s − 0.951·25-s − 0.192i·27-s − 0.0607i·29-s − 0.433·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9447636683\)
\(L(\frac12)\) \(\approx\) \(0.9447636683\)
\(L(\frac{5}{2})\) 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 \( 1 - 7T \)
good5 \( 1 - 15.6iT - 125T^{2} \)
11 \( 1 - 13.3iT - 1.33e3T^{2} \)
13 \( 1 - 7.74iT - 2.19e3T^{2} \)
17 \( 1 + 25.3T + 4.91e3T^{2} \)
19 \( 1 - 71.9iT - 6.85e3T^{2} \)
23 \( 1 - 23.6T + 1.21e4T^{2} \)
29 \( 1 + 9.49iT - 2.43e4T^{2} \)
31 \( 1 + 74.7T + 2.97e4T^{2} \)
37 \( 1 - 34.2iT - 5.06e4T^{2} \)
41 \( 1 + 406.T + 6.89e4T^{2} \)
43 \( 1 - 141. iT - 7.95e4T^{2} \)
47 \( 1 + 127.T + 1.03e5T^{2} \)
53 \( 1 - 106. iT - 1.48e5T^{2} \)
59 \( 1 - 65.5iT - 2.05e5T^{2} \)
61 \( 1 + 35.5iT - 2.26e5T^{2} \)
67 \( 1 - 606. iT - 3.00e5T^{2} \)
71 \( 1 + 921.T + 3.57e5T^{2} \)
73 \( 1 - 705.T + 3.89e5T^{2} \)
79 \( 1 + 294.T + 4.93e5T^{2} \)
83 \( 1 - 291. iT - 5.71e5T^{2} \)
89 \( 1 - 0.646T + 7.04e5T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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

−10.01477214679026705317255829879, −9.006867840727638524750338536237, −8.101969030642893426160201417871, −7.23317764722763030807134483560, −6.53244012085382169780802096527, −5.62544240726412019249787572871, −4.59097693090137656276220144988, −3.65577565925420747340675346844, −2.81325610164003440963634247948, −1.72883195677385531507870849953, 0.22090101780678038292880617367, 1.15110464409834105358944627240, 2.14536574620839846543938455190, 3.48603519286071700621786792810, 4.72150707492854106033020333028, 5.21615893770273885246334462492, 6.22721994051193655593945167249, 7.18575610604368592562453687374, 8.059617063799959016839717398894, 8.744091567682190329952015186612

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