Properties

Label 2-1344-8.5-c3-0-44
Degree $2$
Conductor $1344$
Sign $0.258 + 0.965i$
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 − 6.85i·5-s + 7·7-s − 9·9-s + 30.4i·11-s + 45.1i·13-s − 20.5·15-s − 99.4·17-s − 33.9i·19-s − 21i·21-s + 108.·23-s + 78.0·25-s + 27i·27-s − 187. i·29-s − 174.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.613i·5-s + 0.377·7-s − 0.333·9-s + 0.834i·11-s + 0.962i·13-s − 0.354·15-s − 1.41·17-s − 0.410i·19-s − 0.218i·21-s + 0.982·23-s + 0.624·25-s + 0.192i·27-s − 1.20i·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.258 + 0.965i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.927805606\)
\(L(\frac12)\) \(\approx\) \(1.927805606\)
\(L(\frac{5}{2})\) 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 \( 1 + 6.85iT - 125T^{2} \)
11 \( 1 - 30.4iT - 1.33e3T^{2} \)
13 \( 1 - 45.1iT - 2.19e3T^{2} \)
17 \( 1 + 99.4T + 4.91e3T^{2} \)
19 \( 1 + 33.9iT - 6.85e3T^{2} \)
23 \( 1 - 108.T + 1.21e4T^{2} \)
29 \( 1 + 187. iT - 2.43e4T^{2} \)
31 \( 1 + 174.T + 2.97e4T^{2} \)
37 \( 1 - 241. iT - 5.06e4T^{2} \)
41 \( 1 - 474.T + 6.89e4T^{2} \)
43 \( 1 + 480. iT - 7.95e4T^{2} \)
47 \( 1 - 516.T + 1.03e5T^{2} \)
53 \( 1 - 179. iT - 1.48e5T^{2} \)
59 \( 1 - 22.3iT - 2.05e5T^{2} \)
61 \( 1 + 932. iT - 2.26e5T^{2} \)
67 \( 1 - 665. iT - 3.00e5T^{2} \)
71 \( 1 + 129.T + 3.57e5T^{2} \)
73 \( 1 - 1.08e3T + 3.89e5T^{2} \)
79 \( 1 + 739.T + 4.93e5T^{2} \)
83 \( 1 - 81.6iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 610.T + 9.12e5T^{2} \)
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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

−8.993552056101652830425866950199, −8.393229869147668893649954289945, −7.22524229690237552307417125646, −6.88111374163034795061045688466, −5.74444824904087199906469289330, −4.72232126049197323620670616425, −4.17035251774853256981987277880, −2.53367083635642573388743200486, −1.75884235637082867770877245526, −0.56190256746990078347186960068, 0.873075394083658289056431995214, 2.47220700520306245847670880816, 3.26756290272134245391491393072, 4.24935026080331416227501788290, 5.25534350900019815195519385370, 5.99283100815000402222152495621, 6.98955897215727176342851246356, 7.79931939812896482603210880513, 8.811245141642482113655227877027, 9.216790416968112413947424332014

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