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Properties

Label 2-1332-148.67-c0-0-0
Degree $2$
Conductor $1332$
Sign $0.165 + 0.986i$
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.85 − 0.326i)5-s + (0.866 + 0.500i)8-s + (−0.939 − 1.62i)10-s + (−1.11 − 1.32i)13-s + (0.173 − 0.984i)16-s + (−0.223 + 0.266i)17-s + (−1.20 + 1.43i)20-s + (2.37 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (1.32 + 0.766i)29-s + (−0.984 + 0.173i)32-s + (0.326 + 0.118i)34-s + (−0.766 + 0.642i)37-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (1.85 − 0.326i)5-s + (0.866 + 0.500i)8-s + (−0.939 − 1.62i)10-s + (−1.11 − 1.32i)13-s + (0.173 − 0.984i)16-s + (−0.223 + 0.266i)17-s + (−1.20 + 1.43i)20-s + (2.37 − 0.866i)25-s + (−0.866 + 1.5i)26-s + (1.32 + 0.766i)29-s + (−0.984 + 0.173i)32-s + (0.326 + 0.118i)34-s + (−0.766 + 0.642i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (955, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ 0.165 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120226884\)
\(L(\frac12)\) \(\approx\) \(1.120226884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 \)
37 \( 1 + (0.766 - 0.642i)T \)
good5 \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
41 \( 1 + (-1.50 + 1.26i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.300 - 1.70i)T + (-0.939 - 0.342i)T^{2} \)
59 \( 1 + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.826 + 0.984i)T + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (0.939 - 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (1.11 - 0.642i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.711495129028565444325372926752, −9.125954577152765826459937964749, −8.340466529275402217937170745385, −7.33682139493869176408948054488, −6.16505334215271097162321452910, −5.27907241930703673834405014677, −4.65446117706914957695090281278, −3.04852952161601770219063159311, −2.36439287260751885014870047146, −1.24404624598771149896843958942, 1.63821338709293107296953715705, 2.62319258197596189405011860772, 4.44488306622822276908806842747, 5.15046387891115202152039021574, 6.09369291918259737644122333065, 6.63382691298596714476923597857, 7.32263898724958685795112335475, 8.504243170977355564834340026565, 9.358207018386588814497372900770, 9.717363340882412989217334763263

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