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Properties

Label 2-1344-168.131-c0-0-2
Degree $2$
Conductor $1344$
Sign $0.925 + 0.378i$
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 + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s − 0.999i·15-s + (0.866 + 0.499i)21-s + 0.999·27-s + 1.73·29-s + (0.866 + 1.5i)31-s + (−1.5 − 0.866i)33-s − 0.999·35-s + (−0.866 + 0.499i)45-s + (0.499 − 0.866i)49-s + (−0.866 − 1.5i)53-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)7-s + (−0.499 + 0.866i)9-s + (1.5 − 0.866i)11-s − 0.999i·15-s + (0.866 + 0.499i)21-s + 0.999·27-s + 1.73·29-s + (0.866 + 1.5i)31-s + (−1.5 − 0.866i)33-s − 0.999·35-s + (−0.866 + 0.499i)45-s + (0.499 − 0.866i)49-s + (−0.866 − 1.5i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ 0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044214089\)
\(L(\frac12)\) \(\approx\) \(1.044214089\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.823137834908481792362217650005, −8.888272078858790293307466411725, −8.255839450909219090557205334374, −6.84981600024654094941238193861, −6.45325741592025682828416398580, −6.01276578724309823367846211043, −4.93348118778221201307433922330, −3.40609680580640693071923220201, −2.52062037045688494729184217297, −1.25848184039190432649435190240, 1.25082044945388021995691029981, 2.88296483511209979798052071448, 4.13210875631869480767568490692, 4.57618398507804609854024996548, 5.85073235565921517325116776231, 6.33859514564893309427820824310, 7.17086714498014046189975702599, 8.583350742468628610454519509740, 9.422917479316564245993633253371, 9.733351940414765177316668480991

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