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Properties

Label 2-2144-536.315-c0-0-0
Degree $2$
Conductor $2144$
Sign $0.999 + 0.0389i$
Analytic cond. $1.06999$
Root an. cond. $1.03440$
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  + (−0.0930 + 0.647i)3-s + (0.548 + 0.161i)9-s + (0.759 − 1.06i)11-s + (−0.0845 − 1.77i)17-s + (0.469 − 1.93i)19-s + (−0.654 + 0.755i)25-s + (−0.427 + 0.935i)27-s + (0.620 + 0.591i)33-s + (1.70 + 0.879i)41-s + (−0.975 + 0.627i)43-s + (−0.888 + 0.458i)49-s + (1.15 + 0.110i)51-s + (1.20 + 0.484i)57-s + (1.28 + 1.48i)59-s + (−0.415 − 0.909i)67-s + ⋯
L(s)  = 1  + (−0.0930 + 0.647i)3-s + (0.548 + 0.161i)9-s + (0.759 − 1.06i)11-s + (−0.0845 − 1.77i)17-s + (0.469 − 1.93i)19-s + (−0.654 + 0.755i)25-s + (−0.427 + 0.935i)27-s + (0.620 + 0.591i)33-s + (1.70 + 0.879i)41-s + (−0.975 + 0.627i)43-s + (−0.888 + 0.458i)49-s + (1.15 + 0.110i)51-s + (1.20 + 0.484i)57-s + (1.28 + 1.48i)59-s + (−0.415 − 0.909i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $0.999 + 0.0389i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ 0.999 + 0.0389i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.244167363\)
\(L(\frac12)\) \(\approx\) \(1.244167363\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
67 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (0.0930 - 0.647i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.888 - 0.458i)T^{2} \)
11 \( 1 + (-0.759 + 1.06i)T + (-0.327 - 0.945i)T^{2} \)
13 \( 1 + (0.786 - 0.618i)T^{2} \)
17 \( 1 + (0.0845 + 1.77i)T + (-0.995 + 0.0950i)T^{2} \)
19 \( 1 + (-0.469 + 1.93i)T + (-0.888 - 0.458i)T^{2} \)
23 \( 1 + (-0.235 - 0.971i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.786 + 0.618i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.70 - 0.879i)T + (0.580 + 0.814i)T^{2} \)
43 \( 1 + (0.975 - 0.627i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.723 + 0.690i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-1.28 - 1.48i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.327 - 0.945i)T^{2} \)
71 \( 1 + (0.995 + 0.0950i)T^{2} \)
73 \( 1 + (-0.0552 - 0.0775i)T + (-0.327 + 0.945i)T^{2} \)
79 \( 1 + (-0.928 - 0.371i)T^{2} \)
83 \( 1 + (-0.827 - 0.0789i)T + (0.981 + 0.189i)T^{2} \)
89 \( 1 + (0.205 + 1.43i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (0.0475 - 0.0824i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.396978546102676616956544113862, −8.756164896672872911033494560250, −7.57837057447259804674427596870, −7.02140540238051061567402977806, −6.08567813990726885749050489004, −5.07442340854671377415139354199, −4.54966891977032285178354528476, −3.48820140365297935051649744811, −2.66513766228737090289002989945, −1.03368988018412889348283559718, 1.47131652505479260457091375265, 2.04824753017584057209515181628, 3.81749956086267052974073983170, 4.10861486070622768712470891410, 5.48508343530790898678943617921, 6.30510030101037918520381480879, 6.84640819589663796226444838557, 7.82016524320932145091269406565, 8.221613915049051558629788384105, 9.390987185041718259741364368957

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