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Properties

Label 2-1344-168.59-c0-0-1
Degree $2$
Conductor $1344$
Sign $-0.378 - 0.925i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.378 - 0.925i$
Analytic conductor: \(0.670743\)
Root analytic conductor: \(0.818989\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :0),\ -0.378 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8372625409\)
\(L(\frac12)\) \(\approx\) \(0.8372625409\)
\(L(1)\) 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

−10.09628516631406958477676546234, −9.194573557610966282679063275731, −8.696122980681010076898473186697, −7.50283011673404914184153021165, −6.86523655630003298610788140342, −5.79408313343819392687186257692, −4.90836761345497317000383223653, −4.06317728236256829546983418879, −3.40506527239624254776131140776, −1.75004490403652246042826573893, 0.813594120730090759566796395399, 1.91678987484341886426102484621, 3.67775473624473684262250142631, 4.35960893643869532432205581809, 5.48621514494713523507329074449, 6.26069796160703279339014041498, 7.30574648374265022177096709878, 7.79470319186571949788561361492, 8.590334036500512038562126872631, 9.324148907326950827615494377885

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