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Properties

Label 2-1332-37.18-c0-0-0
Degree $2$
Conductor $1332$
Sign $-0.566 + 0.823i$
Analytic cond. $0.664754$
Root an. cond. $0.815324$
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  + (−1.85 − 0.673i)7-s + (−0.939 − 0.657i)13-s + (0.168 − 1.92i)19-s + (−0.642 + 0.766i)25-s + (−1.28 − 1.28i)31-s + (0.5 − 0.866i)37-s + (−0.123 + 0.123i)43-s + (2.20 + 1.85i)49-s + (−0.811 + 1.15i)61-s + (−0.524 + 1.43i)67-s − 1.53i·73-s + (1.80 − 0.842i)79-s + (1.29 + 1.85i)91-s + (−0.168 − 0.0451i)97-s + (−0.5 + 0.133i)103-s + ⋯
L(s)  = 1  + (−1.85 − 0.673i)7-s + (−0.939 − 0.657i)13-s + (0.168 − 1.92i)19-s + (−0.642 + 0.766i)25-s + (−1.28 − 1.28i)31-s + (0.5 − 0.866i)37-s + (−0.123 + 0.123i)43-s + (2.20 + 1.85i)49-s + (−0.811 + 1.15i)61-s + (−0.524 + 1.43i)67-s − 1.53i·73-s + (1.80 − 0.842i)79-s + (1.29 + 1.85i)91-s + (−0.168 − 0.0451i)97-s + (−0.5 + 0.133i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $-0.566 + 0.823i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ -0.566 + 0.823i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5049873551\)
\(L(\frac12)\) \(\approx\) \(0.5049873551\)
\(L(1)\) 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 \( 1 \)
37 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.642 - 0.766i)T^{2} \)
7 \( 1 + (1.85 + 0.673i)T + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.939 + 0.657i)T + (0.342 + 0.939i)T^{2} \)
17 \( 1 + (-0.342 + 0.939i)T^{2} \)
19 \( 1 + (-0.168 + 1.92i)T + (-0.984 - 0.173i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T^{2} \)
31 \( 1 + (1.28 + 1.28i)T + iT^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.123 - 0.123i)T - iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.642 - 0.766i)T^{2} \)
61 \( 1 + (0.811 - 1.15i)T + (-0.342 - 0.939i)T^{2} \)
67 \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.173 - 0.984i)T^{2} \)
73 \( 1 + 1.53iT - T^{2} \)
79 \( 1 + (-1.80 + 0.842i)T + (0.642 - 0.766i)T^{2} \)
83 \( 1 + (-0.939 - 0.342i)T^{2} \)
89 \( 1 + (0.642 + 0.766i)T^{2} \)
97 \( 1 + (0.168 + 0.0451i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.392260685012751961610570287090, −9.170337697250141729081520463292, −7.55325936776091187030723233446, −7.23561438585920548687584866610, −6.29262214612466102001980359763, −5.44811283756963351702039574318, −4.28573659152661014041654191763, −3.32841181680964320142224985181, −2.51177254433627628489867784321, −0.39427750236906976356026398143, 2.00522407425586386852916414988, 3.12801713952591114319004930112, 3.89485036267912577302139568733, 5.20577793816332326247710498479, 6.10504159341156488054598197598, 6.66089459558415994355222556242, 7.61995113886796291216933857644, 8.598937055363214273633810543706, 9.549481777389093294177599514786, 9.812068523232340843838557427239

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