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Properties

Label 2-1344-8.5-c3-0-10
Degree $2$
Conductor $1344$
Sign $0.707 - 0.707i$
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 − 13.9i·5-s + 7·7-s − 9·9-s + 44.5i·11-s − 81.2i·13-s − 41.7·15-s − 136.·17-s + 114. i·19-s − 21i·21-s + 67.0·23-s − 69.1·25-s + 27i·27-s + 222. i·29-s + 135.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.24i·5-s + 0.377·7-s − 0.333·9-s + 1.22i·11-s − 1.73i·13-s − 0.719·15-s − 1.95·17-s + 1.38i·19-s − 0.218i·21-s + 0.607·23-s − 0.552·25-s + 0.192i·27-s + 1.42i·29-s + 0.783·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.101030077\)
\(L(\frac12)\) \(\approx\) \(1.101030077\)
\(L(\frac{5}{2})\) 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 \( 1 + 13.9iT - 125T^{2} \)
11 \( 1 - 44.5iT - 1.33e3T^{2} \)
13 \( 1 + 81.2iT - 2.19e3T^{2} \)
17 \( 1 + 136.T + 4.91e3T^{2} \)
19 \( 1 - 114. iT - 6.85e3T^{2} \)
23 \( 1 - 67.0T + 1.21e4T^{2} \)
29 \( 1 - 222. iT - 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 - 298. iT - 5.06e4T^{2} \)
41 \( 1 - 88.8T + 6.89e4T^{2} \)
43 \( 1 - 241. iT - 7.95e4T^{2} \)
47 \( 1 + 377.T + 1.03e5T^{2} \)
53 \( 1 - 174. iT - 1.48e5T^{2} \)
59 \( 1 + 26.8iT - 2.05e5T^{2} \)
61 \( 1 + 51.2iT - 2.26e5T^{2} \)
67 \( 1 + 1.01e3iT - 3.00e5T^{2} \)
71 \( 1 + 502.T + 3.57e5T^{2} \)
73 \( 1 + 709.T + 3.89e5T^{2} \)
79 \( 1 + 716.T + 4.93e5T^{2} \)
83 \( 1 - 882. iT - 5.71e5T^{2} \)
89 \( 1 - 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 386.T + 9.12e5T^{2} \)
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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

−9.176032829987779940212013839662, −8.395813371264204246168715011569, −7.916012812734137936658060654178, −6.96076072895445087401349217281, −6.00977132492746353606934824588, −4.95272445075484389035264817023, −4.58180925487412965996083935842, −3.09998932818496412143555074587, −1.84640504921589164662751835853, −1.04150475599401310023662142440, 0.27028106391416598070238583376, 2.15290106472040428084577248350, 2.88200096537152070158847244779, 4.06819452427401389676801926169, 4.67147419853560881523051873498, 6.00414863899231111292300564556, 6.67861391771658959666751461831, 7.28689567599831272332380267313, 8.739510994845275331012160496025, 8.914321419006594824751613619530

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