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Properties

Label 2-2144-536.323-c0-0-0
Degree $2$
Conductor $2144$
Sign $0.714 + 0.699i$
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  + (0.759 − 0.876i)3-s + (−0.0492 − 0.342i)9-s + (0.738 + 0.380i)11-s + (0.341 − 0.325i)17-s + (0.0748 + 0.0588i)19-s + (0.415 − 0.909i)25-s + (0.638 + 0.410i)27-s + (0.894 − 0.358i)33-s + (−0.0671 − 0.276i)41-s + (−1.70 − 0.500i)43-s + (0.235 − 0.971i)49-s + (−0.0260 − 0.546i)51-s + (0.108 − 0.0208i)57-s + (0.827 + 1.81i)59-s + (−0.841 + 0.540i)67-s + ⋯
L(s)  = 1  + (0.759 − 0.876i)3-s + (−0.0492 − 0.342i)9-s + (0.738 + 0.380i)11-s + (0.341 − 0.325i)17-s + (0.0748 + 0.0588i)19-s + (0.415 − 0.909i)25-s + (0.638 + 0.410i)27-s + (0.894 − 0.358i)33-s + (−0.0671 − 0.276i)41-s + (−1.70 − 0.500i)43-s + (0.235 − 0.971i)49-s + (−0.0260 − 0.546i)51-s + (0.108 − 0.0208i)57-s + (0.827 + 1.81i)59-s + (−0.841 + 0.540i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2144\)    =    \(2^{5} \cdot 67\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(1.06999\)
Root analytic conductor: \(1.03440\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2144} (591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2144,\ (\ :0),\ 0.714 + 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599628938\)
\(L(\frac12)\) \(\approx\) \(1.599628938\)
\(L(1)\) 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 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (-0.759 + 0.876i)T + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (-0.235 + 0.971i)T^{2} \)
11 \( 1 + (-0.738 - 0.380i)T + (0.580 + 0.814i)T^{2} \)
13 \( 1 + (0.327 - 0.945i)T^{2} \)
17 \( 1 + (-0.341 + 0.325i)T + (0.0475 - 0.998i)T^{2} \)
19 \( 1 + (-0.0748 - 0.0588i)T + (0.235 + 0.971i)T^{2} \)
23 \( 1 + (0.786 - 0.618i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.327 + 0.945i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.0671 + 0.276i)T + (-0.888 + 0.458i)T^{2} \)
43 \( 1 + (1.70 + 0.500i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (-0.928 - 0.371i)T^{2} \)
53 \( 1 + (-0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.827 - 1.81i)T + (-0.654 + 0.755i)T^{2} \)
61 \( 1 + (-0.580 + 0.814i)T^{2} \)
71 \( 1 + (-0.0475 - 0.998i)T^{2} \)
73 \( 1 + (1.28 - 0.663i)T + (0.580 - 0.814i)T^{2} \)
79 \( 1 + (-0.981 + 0.189i)T^{2} \)
83 \( 1 + (0.0800 + 1.68i)T + (-0.995 + 0.0950i)T^{2} \)
89 \( 1 + (1.21 + 1.40i)T + (-0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.815813910669860219963692666552, −8.544628056958784674098032186497, −7.50868632339319171580798437386, −7.05177577088149232425442390513, −6.25845729871217163454281152985, −5.18896945668989684315821434475, −4.21063102365070962604240421613, −3.17203499642854681865701669796, −2.25921558657502144232825889935, −1.28589296870716277583162449117, 1.47247650982367032183725522716, 2.92703249704584915276568756203, 3.55320790348278714337242828425, 4.35143367461437486540298537294, 5.24738830227461936182962807695, 6.25449264978519829944643003455, 7.03557312051768276665819251408, 8.112463652136065648522774486835, 8.648538938722621422724503525935, 9.445543288589866528255439371061

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