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Properties

Label 2-1344-8.5-c3-0-57
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 − 16.7i·5-s − 7·7-s − 9·9-s + 16.7i·11-s − 86.0i·13-s + 50.3·15-s + 87.5·17-s + 104. i·19-s − 21i·21-s + 171.·23-s − 157.·25-s − 27i·27-s − 88.8i·29-s − 53.1·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.50i·5-s − 0.377·7-s − 0.333·9-s + 0.459i·11-s − 1.83i·13-s + 0.867·15-s + 1.24·17-s + 1.26i·19-s − 0.218i·21-s + 1.55·23-s − 1.25·25-s − 0.192i·27-s − 0.568i·29-s − 0.307·31-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

\(L(2)\) \(\approx\) \(1.664900079\)
\(L(\frac12)\) \(\approx\) \(1.664900079\)
\(L(\frac{5}{2})\) 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 \( 1 + 16.7iT - 125T^{2} \)
11 \( 1 - 16.7iT - 1.33e3T^{2} \)
13 \( 1 + 86.0iT - 2.19e3T^{2} \)
17 \( 1 - 87.5T + 4.91e3T^{2} \)
19 \( 1 - 104. iT - 6.85e3T^{2} \)
23 \( 1 - 171.T + 1.21e4T^{2} \)
29 \( 1 + 88.8iT - 2.43e4T^{2} \)
31 \( 1 + 53.1T + 2.97e4T^{2} \)
37 \( 1 - 215. iT - 5.06e4T^{2} \)
41 \( 1 - 396.T + 6.89e4T^{2} \)
43 \( 1 - 132. iT - 7.95e4T^{2} \)
47 \( 1 + 325.T + 1.03e5T^{2} \)
53 \( 1 + 674. iT - 1.48e5T^{2} \)
59 \( 1 + 254. iT - 2.05e5T^{2} \)
61 \( 1 + 815. iT - 2.26e5T^{2} \)
67 \( 1 + 616. iT - 3.00e5T^{2} \)
71 \( 1 + 41.3T + 3.57e5T^{2} \)
73 \( 1 + 7.03T + 3.89e5T^{2} \)
79 \( 1 + 418.T + 4.93e5T^{2} \)
83 \( 1 - 93.4iT - 5.71e5T^{2} \)
89 \( 1 + 88.6T + 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−9.131480570634814184140268064973, −8.044614800674029177978986413558, −7.87332719718029601243841272081, −6.30134434693325988271887992257, −5.29445534082052588962997177033, −5.05279764386883324852594431172, −3.81089420748976138400795559972, −3.00364356661859664786729831830, −1.36099999816690009218903934793, −0.43394851908069431818590473662, 1.15146534376428090517321615115, 2.51703676927662937577229180507, 3.11089869254001571994811981099, 4.20442782727774254249221707552, 5.55593201252402938216413281048, 6.41326512851449187707832996872, 7.14183523496767618705695772084, 7.37281188456697302406248572968, 8.848965883110000420607665038852, 9.323534970743295407471347020835

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