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Properties

Label 2-1344-84.47-c0-0-1
Degree $2$
Conductor $1344$
Sign $0.605 - 0.795i$
Analytic cond. $0.670743$
Root an. cond. $0.818989$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−1.49 + 0.866i)39-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + 0.999·57-s + (0.499 + 0.866i)63-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + 1.73i·13-s + (0.5 − 0.866i)19-s + 0.999·21-s + (0.5 + 0.866i)25-s − 0.999·27-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)37-s + (−1.49 + 0.866i)39-s − 1.73i·43-s + (−0.499 − 0.866i)49-s + 0.999·57-s + (0.499 + 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319905652\)
\(L(\frac12)\) \(\approx\) \(1.319905652\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.883361188035373906419441087587, −9.052404586130416601736110748213, −8.609628081706607910401976907680, −7.38454495686700454911791844883, −6.94661736959813500805900863679, −5.54313069208453472758614419187, −4.59252321772901172097261051145, −4.08615738087823309013718195764, −2.98334743462291783556861252951, −1.68917018632820173520831839496, 1.24239805936841830342557562929, 2.56209811985790774024556318593, 3.21550602071566993232049331248, 4.67656229756139373130079230119, 5.78329059058980388254034240801, 6.23852075790183689620468265926, 7.57766102744314241722255790277, 8.025229439064058784395647123513, 8.626113634243279066258617216375, 9.605770795477014076635233566986

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