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Properties

Label 2-1332-37.12-c1-0-3
Degree $2$
Conductor $1332$
Sign $-0.429 - 0.903i$
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  + (0.245 − 0.0892i)5-s + (−2.06 + 0.750i)7-s + (−2.40 − 4.15i)11-s + (−0.355 + 2.01i)13-s + (−0.213 − 1.21i)17-s + (5.98 + 5.02i)19-s + (−2.71 + 4.70i)23-s + (−3.77 + 3.17i)25-s + (2.10 + 3.64i)29-s − 2.24·31-s + (−0.438 + 0.367i)35-s + (5.68 + 2.15i)37-s + (−1.39 + 7.88i)41-s − 8.68·43-s + (−3.17 + 5.50i)47-s + ⋯
L(s)  = 1  + (0.109 − 0.0399i)5-s + (−0.778 + 0.283i)7-s + (−0.723 − 1.25i)11-s + (−0.0987 + 0.559i)13-s + (−0.0518 − 0.294i)17-s + (1.37 + 1.15i)19-s + (−0.566 + 0.982i)23-s + (−0.755 + 0.634i)25-s + (0.390 + 0.676i)29-s − 0.402·31-s + (−0.0740 + 0.0621i)35-s + (0.934 + 0.355i)37-s + (−0.217 + 1.23i)41-s − 1.32·43-s + (−0.463 + 0.802i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.429 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $-0.429 - 0.903i$
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: \((2,\ 1332,\ (\ :1/2),\ -0.429 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7922026912\)
\(L(\frac12)\) \(\approx\) \(0.7922026912\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.68 - 2.15i)T \)
good5 \( 1 + (-0.245 + 0.0892i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (2.06 - 0.750i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.40 + 4.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.355 - 2.01i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.213 + 1.21i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-5.98 - 5.02i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.10 - 3.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
41 \( 1 + (1.39 - 7.88i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 8.68T + 43T^{2} \)
47 \( 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.14 - 1.87i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (13.9 + 5.06i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.582 + 3.30i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.62 + 0.590i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (5.88 + 4.94i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.22T + 73T^{2} \)
79 \( 1 + (-10.8 + 3.94i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-1.67 - 9.47i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-8.92 - 3.24i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.412 + 0.714i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.630723987675784516024403680427, −9.357156263225476926360351754304, −8.129868499802952711503014690317, −7.62597490414699292654624612626, −6.41158057617828638309141933559, −5.79913904651095778360279705648, −4.98804648169354886569691801492, −3.53975307619953557394878145003, −3.01663288763618395481253948157, −1.48054664589223682256735506907, 0.32061277125877477902542403863, 2.14800535826854608563126703548, 3.07768681258751882312028658638, 4.24778027663606379623119927766, 5.11609196370110838189674986520, 6.07776218080445578291510978448, 7.01191691676954372492851133515, 7.60738342602784988577145159030, 8.538733172901150758552483424500, 9.646525584425829443123796524696

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