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Properties

Label 2-1344-8.5-c3-0-48
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 − 17.1i·5-s − 7·7-s − 9·9-s − 2.98i·11-s + 39.8i·13-s + 51.4·15-s + 132.·17-s − 83.2i·19-s − 21i·21-s − 80.0·23-s − 168.·25-s − 27i·27-s + 139. i·29-s + 204.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.53i·5-s − 0.377·7-s − 0.333·9-s − 0.0816i·11-s + 0.850i·13-s + 0.884·15-s + 1.89·17-s − 1.00i·19-s − 0.218i·21-s − 0.725·23-s − 1.34·25-s − 0.192i·27-s + 0.894i·29-s + 1.18·31-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

\(L(2)\) \(\approx\) \(1.775596676\)
\(L(\frac12)\) \(\approx\) \(1.775596676\)
\(L(\frac{5}{2})\) 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 \( 1 + 17.1iT - 125T^{2} \)
11 \( 1 + 2.98iT - 1.33e3T^{2} \)
13 \( 1 - 39.8iT - 2.19e3T^{2} \)
17 \( 1 - 132.T + 4.91e3T^{2} \)
19 \( 1 + 83.2iT - 6.85e3T^{2} \)
23 \( 1 + 80.0T + 1.21e4T^{2} \)
29 \( 1 - 139. iT - 2.43e4T^{2} \)
31 \( 1 - 204.T + 2.97e4T^{2} \)
37 \( 1 - 377. iT - 5.06e4T^{2} \)
41 \( 1 - 45.3T + 6.89e4T^{2} \)
43 \( 1 + 540. iT - 7.95e4T^{2} \)
47 \( 1 - 362.T + 1.03e5T^{2} \)
53 \( 1 - 668. iT - 1.48e5T^{2} \)
59 \( 1 + 757. iT - 2.05e5T^{2} \)
61 \( 1 + 690. iT - 2.26e5T^{2} \)
67 \( 1 + 370. iT - 3.00e5T^{2} \)
71 \( 1 + 652.T + 3.57e5T^{2} \)
73 \( 1 - 229.T + 3.89e5T^{2} \)
79 \( 1 - 43.3T + 4.93e5T^{2} \)
83 \( 1 + 802. iT - 5.71e5T^{2} \)
89 \( 1 + 868.T + 7.04e5T^{2} \)
97 \( 1 + 1.18e3T + 9.12e5T^{2} \)
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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

−9.088854656582774170106332377229, −8.486505016491912316668390532966, −7.63521741260434991501404810149, −6.47629282180262953107578319221, −5.49234369427858901369905060150, −4.85609097524401892961065032312, −4.05655890040194156952577960012, −3.02025800301794692674266745240, −1.51999507417849106231711419204, −0.49084618365171616896332860200, 0.992050516482754139942058737944, 2.42840815127270653363021752563, 3.12483783024921526241978249904, 3.98584923646895662748600081674, 5.79067292162892053440424857447, 5.94457977440200265108513611721, 7.08515020041437966622867128855, 7.67683968959839365073669073898, 8.268697325309468805551198383207, 9.786212609528743271533086946792

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