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Properties

Label 2-1344-8.5-c3-0-43
Degree $2$
Conductor $1344$
Sign $0.965 + 0.258i$
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 − 1.12i·5-s + 7·7-s − 9·9-s − 16.9i·11-s − 11.6i·13-s + 3.37·15-s − 69.2·17-s + 87.3i·19-s + 21i·21-s − 49.5·23-s + 123.·25-s − 27i·27-s − 178. i·29-s + 147.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.100i·5-s + 0.377·7-s − 0.333·9-s − 0.464i·11-s − 0.248i·13-s + 0.0581·15-s − 0.988·17-s + 1.05i·19-s + 0.218i·21-s − 0.449·23-s + 0.989·25-s − 0.192i·27-s − 1.14i·29-s + 0.853·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.965 + 0.258i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.898400386\)
\(L(\frac12)\) \(\approx\) \(1.898400386\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.12iT - 125T^{2} \)
11 \( 1 + 16.9iT - 1.33e3T^{2} \)
13 \( 1 + 11.6iT - 2.19e3T^{2} \)
17 \( 1 + 69.2T + 4.91e3T^{2} \)
19 \( 1 - 87.3iT - 6.85e3T^{2} \)
23 \( 1 + 49.5T + 1.21e4T^{2} \)
29 \( 1 + 178. iT - 2.43e4T^{2} \)
31 \( 1 - 147.T + 2.97e4T^{2} \)
37 \( 1 - 241. iT - 5.06e4T^{2} \)
41 \( 1 + 61.3T + 6.89e4T^{2} \)
43 \( 1 + 283. iT - 7.95e4T^{2} \)
47 \( 1 - 300.T + 1.03e5T^{2} \)
53 \( 1 + 44.4iT - 1.48e5T^{2} \)
59 \( 1 + 260. iT - 2.05e5T^{2} \)
61 \( 1 + 151. iT - 2.26e5T^{2} \)
67 \( 1 + 635. iT - 3.00e5T^{2} \)
71 \( 1 - 51.2T + 3.57e5T^{2} \)
73 \( 1 + 346.T + 3.89e5T^{2} \)
79 \( 1 + 173.T + 4.93e5T^{2} \)
83 \( 1 + 524. iT - 5.71e5T^{2} \)
89 \( 1 - 599.T + 7.04e5T^{2} \)
97 \( 1 - 468.T + 9.12e5T^{2} \)
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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

−9.152416461445826460463503745612, −8.437058974582173859637354154638, −7.81296243497059295075665456809, −6.60807123196497196879606870581, −5.85236199926322349608195827828, −4.89233773782111563414829358069, −4.13595505830910846542356882496, −3.13087045478383303890391786068, −1.98170737647934863962856090355, −0.55745344307981047197729562256, 0.852373464144603933813995184472, 2.03068286042105940154204521425, 2.89741832533801341564269262496, 4.27884509695826409634490562382, 4.99949783488290981762812043673, 6.11320316459440287180895594613, 6.95698461075892449730403864481, 7.45101266551721821896953857239, 8.615335424437404255255344390545, 9.005296747309893297311410022385

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