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Properties

Label 2-1944-1944.211-c0-0-0
Degree $2$
Conductor $1944$
Sign $0.971 - 0.236i$
Analytic cond. $0.970182$
Root an. cond. $0.984978$
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  + (0.0968 − 0.995i)2-s + (−0.999 − 0.0387i)3-s + (−0.981 − 0.192i)4-s + (−0.135 + 0.990i)6-s + (−0.286 + 0.957i)8-s + (0.996 + 0.0774i)9-s + (−0.479 − 0.435i)11-s + (0.973 + 0.230i)12-s + (0.925 + 0.378i)16-s + (−0.0830 + 1.42i)17-s + (0.173 − 0.984i)18-s + (−1.59 + 1.04i)19-s + (−0.479 + 0.435i)22-s + (0.323 − 0.946i)24-s + (0.249 + 0.968i)25-s + ⋯
L(s)  = 1  + (0.0968 − 0.995i)2-s + (−0.999 − 0.0387i)3-s + (−0.981 − 0.192i)4-s + (−0.135 + 0.990i)6-s + (−0.286 + 0.957i)8-s + (0.996 + 0.0774i)9-s + (−0.479 − 0.435i)11-s + (0.973 + 0.230i)12-s + (0.925 + 0.378i)16-s + (−0.0830 + 1.42i)17-s + (0.173 − 0.984i)18-s + (−1.59 + 1.04i)19-s + (−0.479 + 0.435i)22-s + (0.323 − 0.946i)24-s + (0.249 + 0.968i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.971 - 0.236i$
Analytic conductor: \(0.970182\)
Root analytic conductor: \(0.984978\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1944,\ (\ :0),\ 0.971 - 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5408365660\)
\(L(\frac12)\) \(\approx\) \(0.5408365660\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0968 + 0.995i)T \)
3 \( 1 + (0.999 + 0.0387i)T \)
good5 \( 1 + (-0.249 - 0.968i)T^{2} \)
7 \( 1 + (-0.533 - 0.845i)T^{2} \)
11 \( 1 + (0.479 + 0.435i)T + (0.0968 + 0.995i)T^{2} \)
13 \( 1 + (-0.0193 - 0.999i)T^{2} \)
17 \( 1 + (0.0830 - 1.42i)T + (-0.993 - 0.116i)T^{2} \)
19 \( 1 + (1.59 - 1.04i)T + (0.396 - 0.918i)T^{2} \)
23 \( 1 + (0.999 + 0.0387i)T^{2} \)
29 \( 1 + (0.627 + 0.778i)T^{2} \)
31 \( 1 + (0.740 + 0.672i)T^{2} \)
37 \( 1 + (-0.973 + 0.230i)T^{2} \)
41 \( 1 + (0.970 + 0.441i)T + (0.657 + 0.753i)T^{2} \)
43 \( 1 + (-1.81 - 0.0705i)T + (0.996 + 0.0774i)T^{2} \)
47 \( 1 + (0.740 - 0.672i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (-1.76 - 0.565i)T + (0.813 + 0.581i)T^{2} \)
61 \( 1 + (-0.925 + 0.378i)T^{2} \)
67 \( 1 + (-0.755 - 1.57i)T + (-0.627 + 0.778i)T^{2} \)
71 \( 1 + (-0.893 - 0.448i)T^{2} \)
73 \( 1 + (-0.776 - 0.822i)T + (-0.0581 + 0.998i)T^{2} \)
79 \( 1 + (-0.323 - 0.946i)T^{2} \)
83 \( 1 + (0.721 - 0.327i)T + (0.657 - 0.753i)T^{2} \)
89 \( 1 + (1.62 - 0.385i)T + (0.893 - 0.448i)T^{2} \)
97 \( 1 + (-0.569 + 0.441i)T + (0.249 - 0.968i)T^{2} \)
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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

−9.728100300522862588112810729706, −8.661179672632670415810893208251, −8.096794037245591967759723890267, −6.92004221511818222805157518817, −5.85124646961588284433225875138, −5.51158492530765626342341840306, −4.27012762846407812656061525132, −3.80230944839748347670125574203, −2.35232155268568316151073909235, −1.30183047900332135044520001307, 0.47536756092839370101794135007, 2.45800998924197299847690085139, 4.00479121354927351586738668622, 4.78443566731346682234865503741, 5.26477542590915984847397348527, 6.34142386140415642247554566034, 6.82789294405234163174296170412, 7.52865696631819480379944273754, 8.474381445256028241454835382404, 9.290097425150657711586239377900

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