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Properties

Label 2-1332-37.12-c1-0-7
Degree $2$
Conductor $1332$
Sign $0.766 - 0.641i$
Analytic cond. $10.6360$
Root an. cond. $3.26129$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 + 0.613i)5-s + (2.82 − 1.02i)7-s + (1.22 + 2.12i)11-s + (−0.123 + 0.698i)13-s + (0.266 + 1.51i)17-s + (−0.404 − 0.339i)19-s + (0.594 − 1.02i)23-s + (−1.36 + 1.14i)25-s + (0.173 + 0.300i)29-s + 7.53·31-s + (−4.13 + 3.46i)35-s + (−0.665 + 6.04i)37-s + (1.20 − 6.86i)41-s + 3.76·43-s + (−1.56 + 2.70i)47-s + ⋯
L(s)  = 1  + (−0.753 + 0.274i)5-s + (1.06 − 0.388i)7-s + (0.369 + 0.640i)11-s + (−0.0341 + 0.193i)13-s + (0.0647 + 0.366i)17-s + (−0.0928 − 0.0779i)19-s + (0.123 − 0.214i)23-s + (−0.273 + 0.229i)25-s + (0.0322 + 0.0558i)29-s + 1.35·31-s + (−0.698 + 0.586i)35-s + (−0.109 + 0.993i)37-s + (0.188 − 1.07i)41-s + 0.574·43-s + (−0.227 + 0.394i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $0.766 - 0.641i$
Analytic conductor: \(10.6360\)
Root analytic conductor: \(3.26129\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :1/2),\ 0.766 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605277346\)
\(L(\frac12)\) \(\approx\) \(1.605277346\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.665 - 6.04i)T \)
good5 \( 1 + (1.68 - 0.613i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-2.82 + 1.02i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.22 - 2.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.123 - 0.698i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.266 - 1.51i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (0.404 + 0.339i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-0.594 + 1.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.173 - 0.300i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.53T + 31T^{2} \)
41 \( 1 + (-1.20 + 6.86i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 3.76T + 43T^{2} \)
47 \( 1 + (1.56 - 2.70i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.80 - 3.56i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-10.8 - 3.93i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.34 - 7.64i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.01 + 1.82i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.62 - 3.87i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 6.37T + 73T^{2} \)
79 \( 1 + (11.3 - 4.11i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.94 - 11.0i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.57 - 0.936i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (1.94 - 3.36i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.809231632901829585106622961408, −8.722126161170743724457022697607, −8.058437688061811963468128999898, −7.33007259941779734727892096154, −6.63504491415184805045867170589, −5.41324050118075165348727948717, −4.42773446136772462027903778889, −3.88370685571508670339985386621, −2.48504313888771088549759841384, −1.19939663238097430398270335919, 0.802461304236057234486921536106, 2.22751085010694853391328842050, 3.50201475358799647306637396638, 4.44186552524568338678592133432, 5.23290456354233684500240285057, 6.16378982712918265071084571479, 7.23909230988294545282934577657, 8.155462460394806324484342901322, 8.444667224781143684999848407161, 9.432198842815734896876883951978

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