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Properties

Label 2-1344-56.27-c3-0-88
Degree $2$
Conductor $1344$
Sign $-0.968 + 0.249i$
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 + 10.4·5-s + (9.40 − 15.9i)7-s − 9·9-s − 51.5·11-s + 46.4·13-s − 31.2i·15-s + 99.1i·17-s − 12.5i·19-s + (−47.8 − 28.2i)21-s − 129. i·23-s − 16.2·25-s + 27i·27-s − 207. i·29-s − 139.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.932·5-s + (0.507 − 0.861i)7-s − 0.333·9-s − 1.41·11-s + 0.990·13-s − 0.538i·15-s + 1.41i·17-s − 0.151i·19-s + (−0.497 − 0.293i)21-s − 1.17i·23-s − 0.130·25-s + 0.192i·27-s − 1.32i·29-s − 0.808·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.968 + 0.249i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.968 + 0.249i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.368450581\)
\(L(\frac12)\) \(\approx\) \(1.368450581\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.40 + 15.9i)T \)
good5 \( 1 - 10.4T + 125T^{2} \)
11 \( 1 + 51.5T + 1.33e3T^{2} \)
13 \( 1 - 46.4T + 2.19e3T^{2} \)
17 \( 1 - 99.1iT - 4.91e3T^{2} \)
19 \( 1 + 12.5iT - 6.85e3T^{2} \)
23 \( 1 + 129. iT - 1.21e4T^{2} \)
29 \( 1 + 207. iT - 2.43e4T^{2} \)
31 \( 1 + 139.T + 2.97e4T^{2} \)
37 \( 1 - 152. iT - 5.06e4T^{2} \)
41 \( 1 + 317. iT - 6.89e4T^{2} \)
43 \( 1 + 366.T + 7.95e4T^{2} \)
47 \( 1 - 103.T + 1.03e5T^{2} \)
53 \( 1 - 24.2iT - 1.48e5T^{2} \)
59 \( 1 + 339. iT - 2.05e5T^{2} \)
61 \( 1 - 25.3T + 2.26e5T^{2} \)
67 \( 1 + 660.T + 3.00e5T^{2} \)
71 \( 1 + 0.771iT - 3.57e5T^{2} \)
73 \( 1 - 342. iT - 3.89e5T^{2} \)
79 \( 1 + 951. iT - 4.93e5T^{2} \)
83 \( 1 - 244. iT - 5.71e5T^{2} \)
89 \( 1 + 1.17e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.40e3iT - 9.12e5T^{2} \)
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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

−8.578530209106891070908468964889, −8.146290658095522484282000426695, −7.27696582299553601987754658286, −6.27906586587065276701957941685, −5.72880154855635963158774897179, −4.68836025224223008731142202216, −3.60218921005916804075157755962, −2.29667688301768819985632084070, −1.54739032214500982673666255683, −0.28372954238777574462287402305, 1.49763143121297284864406725210, 2.53776536679092207397389681092, 3.38923867607043596658944081648, 4.86674108588964510775677167182, 5.40828649845947572224289674542, 5.96487180928447600496575006694, 7.24315571894120680338645839170, 8.134646776463889930721559279696, 8.978691999130910733039448540785, 9.533861325957654844248161595637

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