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Properties

Label 2-644-161.10-c1-0-5
Degree $2$
Conductor $644$
Sign $0.999 - 0.0381i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (0.623 + 1.20i)3-s + (−3.67 + 0.351i)5-s + (0.914 − 2.48i)7-s + (0.667 − 0.937i)9-s + (4.82 − 1.67i)11-s + (0.293 + 0.999i)13-s + (−2.71 − 4.22i)15-s + (6.56 + 2.62i)17-s + (−7.01 + 2.80i)19-s + (3.57 − 0.441i)21-s + (1.34 − 4.60i)23-s + (8.49 − 1.63i)25-s + (5.58 + 0.803i)27-s + (−0.133 − 0.928i)29-s + (3.61 − 0.171i)31-s + ⋯
L(s)  = 1  + (0.359 + 0.697i)3-s + (−1.64 + 0.157i)5-s + (0.345 − 0.938i)7-s + (0.222 − 0.312i)9-s + (1.45 − 0.503i)11-s + (0.0813 + 0.277i)13-s + (−0.701 − 1.09i)15-s + (1.59 + 0.637i)17-s + (−1.60 + 0.643i)19-s + (0.779 − 0.0962i)21-s + (0.280 − 0.959i)23-s + (1.69 − 0.327i)25-s + (1.07 + 0.154i)27-s + (−0.0248 − 0.172i)29-s + (0.648 − 0.0308i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.999 - 0.0381i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.999 - 0.0381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45867 + 0.0278501i\)
\(L(\frac12)\) \(\approx\) \(1.45867 + 0.0278501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.914 + 2.48i)T \)
23 \( 1 + (-1.34 + 4.60i)T \)
good3 \( 1 + (-0.623 - 1.20i)T + (-1.74 + 2.44i)T^{2} \)
5 \( 1 + (3.67 - 0.351i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-4.82 + 1.67i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.293 - 0.999i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-6.56 - 2.62i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (7.01 - 2.80i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.133 + 0.928i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-3.61 + 0.171i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-5.13 - 3.65i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-2.40 - 1.10i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.463 - 0.720i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-10.5 + 6.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.58 + 10.0i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.07 - 1.23i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (4.45 + 2.29i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.918 - 4.76i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-4.77 - 5.50i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (5.52 - 7.02i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (2.59 - 2.71i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (4.28 + 9.38i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.167 + 3.52i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (1.43 - 3.14i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.56565117357730516113452151974, −9.836760323232073724975961187479, −8.563610989706066208654287477524, −8.196062737830660523965463832205, −7.08872153649672852834471808119, −6.29045206062656194023059029011, −4.39709544339525686666365511997, −4.04170503912070406371034997658, −3.37188217782036158938994328123, −1.00782175138635689381929019595, 1.24233334118439210092835146587, 2.74105505077278310298522941914, 3.98915062717387536852587189922, 4.85076281125819160288511607715, 6.24422276225734339173319057897, 7.37889665907953086361889704559, 7.77811925599184868039609855398, 8.678450995270038765384247866248, 9.392634433859678890283688443078, 10.83442719678895424954580014020

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