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Properties

Label 2-1344-8.5-c3-0-32
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 11.7i·5-s − 7·7-s − 9·9-s + 72.4i·11-s − 50.7i·13-s − 35.2·15-s + 60.4·17-s + 33.8i·19-s + 21i·21-s − 116.·23-s − 13.1·25-s + 27i·27-s − 13.1i·29-s + 250.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.05i·5-s − 0.377·7-s − 0.333·9-s + 1.98i·11-s − 1.08i·13-s − 0.607·15-s + 0.862·17-s + 0.408i·19-s + 0.218i·21-s − 1.05·23-s − 0.105·25-s + 0.192i·27-s − 0.0844i·29-s + 1.44·31-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

\(L(2)\) \(\approx\) \(1.895030742\)
\(L(\frac12)\) \(\approx\) \(1.895030742\)
\(L(\frac{5}{2})\) 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 \( 1 + 11.7iT - 125T^{2} \)
11 \( 1 - 72.4iT - 1.33e3T^{2} \)
13 \( 1 + 50.7iT - 2.19e3T^{2} \)
17 \( 1 - 60.4T + 4.91e3T^{2} \)
19 \( 1 - 33.8iT - 6.85e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 + 13.1iT - 2.43e4T^{2} \)
31 \( 1 - 250.T + 2.97e4T^{2} \)
37 \( 1 - 92.7iT - 5.06e4T^{2} \)
41 \( 1 + 69.1T + 6.89e4T^{2} \)
43 \( 1 - 69.6iT - 7.95e4T^{2} \)
47 \( 1 - 346.T + 1.03e5T^{2} \)
53 \( 1 - 585. iT - 1.48e5T^{2} \)
59 \( 1 - 66.1iT - 2.05e5T^{2} \)
61 \( 1 + 492. iT - 2.26e5T^{2} \)
67 \( 1 - 543. iT - 3.00e5T^{2} \)
71 \( 1 - 365.T + 3.57e5T^{2} \)
73 \( 1 - 374.T + 3.89e5T^{2} \)
79 \( 1 - 670.T + 4.93e5T^{2} \)
83 \( 1 + 595. iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + 218.T + 9.12e5T^{2} \)
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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

−9.179217344724345406956812193066, −8.090993085847127209847266689641, −7.72489241602727498966425193084, −6.72697538095172205253009851990, −5.77323368078882477984953964692, −4.95636044691119473861773287425, −4.11438901302923093839767796475, −2.78916610449068454943706361633, −1.67986222372957237395272185244, −0.70596965120411915090365517412, 0.68223341031286990427632238292, 2.42829458231761548103115243773, 3.31932157154225688693528894149, 3.91939198688195673496808511404, 5.23710331462390592402494352478, 6.18700627279059301063115497468, 6.63046660155275427167148448516, 7.80568987115353115304591660240, 8.607123954674907123004440869974, 9.386909493437063382943942617969

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