Properties

Degree $2$
Conductor $1344$
Sign $0.332 + 0.943i$
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 − 1.64·5-s + (−15.2 + 10.4i)7-s − 9·9-s − 20.3·11-s − 13.0·13-s − 4.92i·15-s + 23.9i·17-s + 87.7i·19-s + (−31.4 − 45.8i)21-s + 73.6i·23-s − 122.·25-s − 27i·27-s + 58.9i·29-s + 124.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.146·5-s + (−0.824 + 0.565i)7-s − 0.333·9-s − 0.557·11-s − 0.279·13-s − 0.0847i·15-s + 0.341i·17-s + 1.05i·19-s + (−0.326 − 0.476i)21-s + 0.667i·23-s − 0.978·25-s − 0.192i·27-s + 0.377i·29-s + 0.723·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.332 + 0.943i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.332 + 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3237040519\)
\(L(\frac12)\) \(\approx\) \(0.3237040519\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.2 - 10.4i)T \)
good5 \( 1 + 1.64T + 125T^{2} \)
11 \( 1 + 20.3T + 1.33e3T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 - 23.9iT - 4.91e3T^{2} \)
19 \( 1 - 87.7iT - 6.85e3T^{2} \)
23 \( 1 - 73.6iT - 1.21e4T^{2} \)
29 \( 1 - 58.9iT - 2.43e4T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + 56.5iT - 5.06e4T^{2} \)
41 \( 1 - 135. iT - 6.89e4T^{2} \)
43 \( 1 + 259.T + 7.95e4T^{2} \)
47 \( 1 + 217.T + 1.03e5T^{2} \)
53 \( 1 + 529. iT - 1.48e5T^{2} \)
59 \( 1 - 685. iT - 2.05e5T^{2} \)
61 \( 1 + 149.T + 2.26e5T^{2} \)
67 \( 1 - 409.T + 3.00e5T^{2} \)
71 \( 1 + 885. iT - 3.57e5T^{2} \)
73 \( 1 + 269. iT - 3.89e5T^{2} \)
79 \( 1 + 902. iT - 4.93e5T^{2} \)
83 \( 1 - 623. iT - 5.71e5T^{2} \)
89 \( 1 + 986. iT - 7.04e5T^{2} \)
97 \( 1 + 179. iT - 9.12e5T^{2} \)
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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

−9.192606585917360753694953053820, −8.316669282568678736496220336336, −7.59853198500983749524350100053, −6.43560641938394770982174934491, −5.74266320942497302173363097590, −4.90182512124760573428540625107, −3.76604809029886815057413059456, −3.06072713080073834821970986089, −1.87368830559007105494122608512, −0.096365748724189139892515272173, 0.78733597964001700386792006559, 2.29920443691057837798301087458, 3.13684202429515561946585230033, 4.26782711119306582079454191723, 5.24231057751442429649960137323, 6.32534770991053083359436629052, 6.92959917227792282846648065993, 7.70272820083288144424598264408, 8.468154030288431088015639917420, 9.490710415081382810663378495513

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