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Properties

Label 2-1344-56.27-c3-0-94
Degree $2$
Conductor $1344$
Sign $-0.382 - 0.924i$
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 + 4.35·5-s + (7.09 − 17.1i)7-s − 9·9-s − 43.1·11-s − 26.5·13-s − 13.0i·15-s − 76.6i·17-s − 58.7i·19-s + (−51.3 − 21.2i)21-s − 0.747i·23-s − 106.·25-s + 27i·27-s + 64.9i·29-s + 165.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.389·5-s + (0.382 − 0.923i)7-s − 0.333·9-s − 1.18·11-s − 0.565·13-s − 0.224i·15-s − 1.09i·17-s − 0.709i·19-s + (−0.533 − 0.221i)21-s − 0.00677i·23-s − 0.848·25-s + 0.192i·27-s + 0.416i·29-s + 0.958·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.382 - 0.924i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.382 - 0.924i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.01139154490\)
\(L(\frac12)\) \(\approx\) \(0.01139154490\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.09 + 17.1i)T \)
good5 \( 1 - 4.35T + 125T^{2} \)
11 \( 1 + 43.1T + 1.33e3T^{2} \)
13 \( 1 + 26.5T + 2.19e3T^{2} \)
17 \( 1 + 76.6iT - 4.91e3T^{2} \)
19 \( 1 + 58.7iT - 6.85e3T^{2} \)
23 \( 1 + 0.747iT - 1.21e4T^{2} \)
29 \( 1 - 64.9iT - 2.43e4T^{2} \)
31 \( 1 - 165.T + 2.97e4T^{2} \)
37 \( 1 - 389. iT - 5.06e4T^{2} \)
41 \( 1 - 290. iT - 6.89e4T^{2} \)
43 \( 1 - 32.4T + 7.95e4T^{2} \)
47 \( 1 + 368.T + 1.03e5T^{2} \)
53 \( 1 + 588. iT - 1.48e5T^{2} \)
59 \( 1 - 471. iT - 2.05e5T^{2} \)
61 \( 1 + 448.T + 2.26e5T^{2} \)
67 \( 1 - 707.T + 3.00e5T^{2} \)
71 \( 1 + 997. iT - 3.57e5T^{2} \)
73 \( 1 - 1.22e3iT - 3.89e5T^{2} \)
79 \( 1 - 160. iT - 4.93e5T^{2} \)
83 \( 1 - 1.00e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 + 729. iT - 9.12e5T^{2} \)
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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

−8.453056442487190726229832665997, −7.81395835857904274469929380944, −7.11641719547626788488218361630, −6.37075562950896401111928306024, −5.14012204975391634621976593594, −4.68837019509649151802930902820, −3.16846768727424789236652182309, −2.32259190399463619698134696141, −1.09958830177507272275201287209, −0.00261063304850998864756706012, 1.87623554452504161422872099035, 2.62329619810462795569509609199, 3.82888748618585617435908027493, 4.87538041045897243311946980997, 5.62679627675058760577597155909, 6.18188830665151322954375068879, 7.64094447271665041606317921688, 8.179650822722805098453868495018, 9.051740339491843096143022495373, 9.837332947402482196765270114489

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