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Properties

Label 2-1344-168.59-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(1.447964396\)
\(L(\frac12)\) \(\approx\) \(1.447964396\)
\(L(1)\) 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

−9.431582520513879569095754467496, −8.623749198175470930078247888345, −8.218948284176846137266759215836, −7.38495220948860389074848444487, −6.26562205379882099368558365486, −5.51452856963098921594943686419, −4.86607239995514020870196075598, −3.17304829934716906974248437885, −2.33876627395982645264829538381, −1.30943084078403990086472268939, 2.05498333434485971264513030392, 2.71064254738686753267134994663, 4.04556870434373230354377839356, 4.91281096068196826600669154374, 5.52712815536586865112860961791, 6.74323885533229220343070846103, 7.80487893253625665380189767429, 8.198365700856281798314647197309, 9.399619969219002912617044879771, 10.03014076204099189335427300800

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