Properties

Degree $2$
Conductor $1344$
Sign $0.406 + 0.913i$
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 + 12.5·5-s + (18.2 − 2.88i)7-s − 9·9-s + 55.4·11-s + 92.1·13-s − 37.5i·15-s − 118. i·17-s − 155. i·19-s + (−8.66 − 54.8i)21-s + 125. i·23-s + 31.6·25-s + 27i·27-s − 131. i·29-s − 66.0·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.11·5-s + (0.987 − 0.155i)7-s − 0.333·9-s + 1.51·11-s + 1.96·13-s − 0.646i·15-s − 1.68i·17-s − 1.87i·19-s + (−0.0900 − 0.570i)21-s + 1.13i·23-s + 0.253·25-s + 0.192i·27-s − 0.842i·29-s − 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.406 + 0.913i$
Motivic weight: \(3\)
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.406 + 0.913i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.881688195\)
\(L(\frac12)\) \(\approx\) \(3.881688195\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.2 + 2.88i)T \)
good5 \( 1 - 12.5T + 125T^{2} \)
11 \( 1 - 55.4T + 1.33e3T^{2} \)
13 \( 1 - 92.1T + 2.19e3T^{2} \)
17 \( 1 + 118. iT - 4.91e3T^{2} \)
19 \( 1 + 155. iT - 6.85e3T^{2} \)
23 \( 1 - 125. iT - 1.21e4T^{2} \)
29 \( 1 + 131. iT - 2.43e4T^{2} \)
31 \( 1 + 66.0T + 2.97e4T^{2} \)
37 \( 1 - 147. iT - 5.06e4T^{2} \)
41 \( 1 + 20.3iT - 6.89e4T^{2} \)
43 \( 1 + 355.T + 7.95e4T^{2} \)
47 \( 1 + 79.5T + 1.03e5T^{2} \)
53 \( 1 - 463. iT - 1.48e5T^{2} \)
59 \( 1 - 580. iT - 2.05e5T^{2} \)
61 \( 1 - 587.T + 2.26e5T^{2} \)
67 \( 1 + 496.T + 3.00e5T^{2} \)
71 \( 1 + 232. iT - 3.57e5T^{2} \)
73 \( 1 - 551. iT - 3.89e5T^{2} \)
79 \( 1 - 437. iT - 4.93e5T^{2} \)
83 \( 1 + 191. iT - 5.71e5T^{2} \)
89 \( 1 + 93.7iT - 7.04e5T^{2} \)
97 \( 1 - 758. iT - 9.12e5T^{2} \)
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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

−9.091527620121302848341469018100, −8.448642194719918332120695996889, −7.28766923212592547665006921634, −6.64229657008665734265944656749, −5.85939757803443469670503887843, −5.01904455863587628658554802998, −3.92245205113131901937446173575, −2.67522256887267913096609204199, −1.51075988532183141874963265052, −0.993865546873576556899756578305, 1.50100015198367664218127040220, 1.71014046658485481046886993298, 3.63535765947389657674283400020, 4.03602542661140418654681815044, 5.36263569958889021368168257620, 6.09140567393966279587335938932, 6.52109720132618743130889278174, 8.279516896585644390813965598056, 8.482938885727058851038782358576, 9.350759261943019966450088895307

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