Properties

Degree $2$
Conductor $1344$
Sign $0.117 - 0.993i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 5.86·5-s + (18.3 − 2.64i)7-s − 9·9-s + 34.5·11-s − 55.8·13-s − 17.5i·15-s − 2.77i·17-s − 67.8i·19-s + (7.94 + 54.9i)21-s + 176. i·23-s − 90.6·25-s − 27i·27-s − 116. i·29-s + 312.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.524·5-s + (0.989 − 0.143i)7-s − 0.333·9-s + 0.946·11-s − 1.19·13-s − 0.302i·15-s − 0.0395i·17-s − 0.819i·19-s + (0.0826 + 0.571i)21-s + 1.59i·23-s − 0.724·25-s − 0.192i·27-s − 0.743i·29-s + 1.80·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.117 - 0.993i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.117 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.834629452\)
\(L(\frac12)\) \(\approx\) \(1.834629452\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 + 2.64i)T \)
good5 \( 1 + 5.86T + 125T^{2} \)
11 \( 1 - 34.5T + 1.33e3T^{2} \)
13 \( 1 + 55.8T + 2.19e3T^{2} \)
17 \( 1 + 2.77iT - 4.91e3T^{2} \)
19 \( 1 + 67.8iT - 6.85e3T^{2} \)
23 \( 1 - 176. iT - 1.21e4T^{2} \)
29 \( 1 + 116. iT - 2.43e4T^{2} \)
31 \( 1 - 312.T + 2.97e4T^{2} \)
37 \( 1 - 118. iT - 5.06e4T^{2} \)
41 \( 1 - 280. iT - 6.89e4T^{2} \)
43 \( 1 - 15.1T + 7.95e4T^{2} \)
47 \( 1 - 6.34T + 1.03e5T^{2} \)
53 \( 1 - 23.2iT - 1.48e5T^{2} \)
59 \( 1 + 288. iT - 2.05e5T^{2} \)
61 \( 1 - 514.T + 2.26e5T^{2} \)
67 \( 1 + 295.T + 3.00e5T^{2} \)
71 \( 1 - 475. iT - 3.57e5T^{2} \)
73 \( 1 + 473. iT - 3.89e5T^{2} \)
79 \( 1 - 796. iT - 4.93e5T^{2} \)
83 \( 1 + 877. iT - 5.71e5T^{2} \)
89 \( 1 + 33.8iT - 7.04e5T^{2} \)
97 \( 1 - 700. iT - 9.12e5T^{2} \)
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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

−9.593386307528508353491340035673, −8.582830426145552111244589157177, −7.84308911046667281620460887601, −7.14915683313182072460416948968, −6.06273896801221161677933773158, −4.93112688340427429571714583008, −4.45452530050322177441632999551, −3.48752480374964706083028127949, −2.28110779337490108525266933572, −0.960505914525455393521701376273, 0.50709343072705565489100658958, 1.68858691174473775594161748404, 2.64521776037386951247826076897, 4.00975818181440843756612838717, 4.73110248145281459140923517565, 5.77018885326944076797230013392, 6.73781263510436757108890351236, 7.46400342568459291873377223909, 8.226155012463908978333595649436, 8.791962055682152712761920109337

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