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Properties

Label 2-2144-536.291-c0-0-0
Degree $2$
Conductor $2144$
Sign $-0.972 - 0.234i$
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  + (0.279 − 1.94i)3-s + (−2.74 − 0.804i)9-s + (−1.30 − 0.124i)11-s + (−0.0845 − 0.0436i)17-s + (−0.839 − 0.800i)19-s + (−0.654 + 0.755i)25-s + (−1.51 + 3.31i)27-s + (−0.606 + 2.49i)33-s + (−0.0913 − 1.91i)41-s + (1.67 − 1.07i)43-s + (0.0475 − 0.998i)49-s + (−0.108 + 0.152i)51-s + (−1.79 + 1.40i)57-s + (−0.428 − 0.494i)59-s + (−0.415 − 0.909i)67-s + ⋯
L(s)  = 1  + (0.279 − 1.94i)3-s + (−2.74 − 0.804i)9-s + (−1.30 − 0.124i)11-s + (−0.0845 − 0.0436i)17-s + (−0.839 − 0.800i)19-s + (−0.654 + 0.755i)25-s + (−1.51 + 3.31i)27-s + (−0.606 + 2.49i)33-s + (−0.0913 − 1.91i)41-s + (1.67 − 1.07i)43-s + (0.0475 − 0.998i)49-s + (−0.108 + 0.152i)51-s + (−1.79 + 1.40i)57-s + (−0.428 − 0.494i)59-s + (−0.415 − 0.909i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $-0.972 - 0.234i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ -0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7618238815\)
\(L(\frac12)\) \(\approx\) \(0.7618238815\)
\(L(1)\) 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 \( 1 + (-0.279 + 1.94i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.0475 + 0.998i)T^{2} \)
11 \( 1 + (1.30 + 0.124i)T + (0.981 + 0.189i)T^{2} \)
13 \( 1 + (-0.928 - 0.371i)T^{2} \)
17 \( 1 + (0.0845 + 0.0436i)T + (0.580 + 0.814i)T^{2} \)
19 \( 1 + (0.839 + 0.800i)T + (0.0475 + 0.998i)T^{2} \)
23 \( 1 + (-0.723 + 0.690i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.0913 + 1.91i)T + (-0.995 + 0.0950i)T^{2} \)
43 \( 1 + (-1.67 + 1.07i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.235 - 0.971i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.428 + 0.494i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.981 + 0.189i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (-1.76 + 0.168i)T + (0.981 - 0.189i)T^{2} \)
79 \( 1 + (0.786 - 0.618i)T^{2} \)
83 \( 1 + (0.481 - 0.676i)T + (-0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.642111292068671449377702174049, −7.908395727961601525659301270261, −7.35579406580391624555955187571, −6.71598836926329108972066435863, −5.82765206803428128493924366946, −5.18839163609820844959087084709, −3.61802494045990232252400340736, −2.52246262183385001763107060601, −1.99600749554856869370528094734, −0.47100302616143910142751063894, 2.39890868790787875174678308932, 3.12039665468600500452153978399, 4.20963975874864302849242332169, 4.62512819802410840164864718451, 5.59719201711912837110160468039, 6.18615516740976677629821716821, 7.82580908361666456950213483919, 8.186624472644861167636068183658, 9.087622300733041720754789258008, 9.785775736272190995152821026536

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