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Properties

Label 2-1332-148.71-c0-0-0
Degree $2$
Conductor $1332$
Sign $0.806 + 0.590i$
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  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (1.43 + 0.524i)5-s + (0.500 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.326 − 1.85i)17-s + (−0.266 + 1.50i)20-s + (1.03 + 0.866i)25-s + (−0.5 + 0.866i)26-s + (0.173 − 0.300i)29-s + (0.939 + 0.342i)32-s + (−1.43 + 1.20i)34-s + (0.173 + 0.984i)37-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (1.43 + 0.524i)5-s + (0.500 − 0.866i)8-s + (−0.766 − 1.32i)10-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)16-s + (0.326 − 1.85i)17-s + (−0.266 + 1.50i)20-s + (1.03 + 0.866i)25-s + (−0.5 + 0.866i)26-s + (0.173 − 0.300i)29-s + (0.939 + 0.342i)32-s + (−1.43 + 1.20i)34-s + (0.173 + 0.984i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.806 + 0.590i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ 0.806 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9617775229\)
\(L(\frac12)\) \(\approx\) \(0.9617775229\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.766 + 0.642i)T \)
3 \( 1 \)
37 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
19 \( 1 + (-0.173 + 0.984i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T^{2} \)
89 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.721771658985979029086576092779, −9.337228279036013462059800039513, −8.220521374061174877045141049818, −7.40510375497308897397730348949, −6.58189595535246395800688311177, −5.65453225742880855369785804137, −4.63307100952548713710141799140, −2.98866261337502159390810470789, −2.64966567447428448359181965692, −1.25964793393972557800305201388, 1.49656749447291047028770304996, 2.19554804761560591217211892929, 4.08702538873636490941929209591, 5.24253378872203004375422178454, 5.88566836977991396860465277419, 6.53522180810120053263703308762, 7.46327320229505030008477125713, 8.569955136039090909595107609490, 8.983415824322096787071132566951, 9.784735647628378559245600661817

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