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Properties

Label 2-1008-63.25-c1-0-29
Degree $2$
Conductor $1008$
Sign $0.997 + 0.0735i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (1.70 + 0.283i)3-s + (0.0731 − 0.126i)5-s + (2.33 − 1.25i)7-s + (2.83 + 0.969i)9-s + (0.832 + 1.44i)11-s + (0.0999 + 0.173i)13-s + (0.160 − 0.195i)15-s + (3.13 − 5.43i)17-s + (−3.45 − 5.99i)19-s + (4.33 − 1.47i)21-s + (−3.09 + 5.35i)23-s + (2.48 + 4.31i)25-s + (4.57 + 2.46i)27-s + (−2.46 + 4.27i)29-s + 2.51·31-s + ⋯
L(s)  = 1  + (0.986 + 0.163i)3-s + (0.0327 − 0.0566i)5-s + (0.880 − 0.473i)7-s + (0.946 + 0.323i)9-s + (0.250 + 0.434i)11-s + (0.0277 + 0.0480i)13-s + (0.0415 − 0.0505i)15-s + (0.760 − 1.31i)17-s + (−0.793 − 1.37i)19-s + (0.946 − 0.322i)21-s + (−0.644 + 1.11i)23-s + (0.497 + 0.862i)25-s + (0.880 + 0.473i)27-s + (−0.458 + 0.793i)29-s + 0.452·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.997 + 0.0735i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.997 + 0.0735i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.581027917\)
\(L(\frac12)\) \(\approx\) \(2.581027917\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.283i)T \)
7 \( 1 + (-2.33 + 1.25i)T \)
good5 \( 1 + (-0.0731 + 0.126i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.832 - 1.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0999 - 0.173i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.13 + 5.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 5.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 - 5.35i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.46 - 4.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.51T + 31T^{2} \)
37 \( 1 + (3.50 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.15 - 2.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.940 + 1.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.81T + 47T^{2} \)
53 \( 1 + (2.67 - 4.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.57T + 59T^{2} \)
61 \( 1 + 0.678T + 61T^{2} \)
67 \( 1 - 6.18T + 67T^{2} \)
71 \( 1 + 1.27T + 71T^{2} \)
73 \( 1 + (0.778 - 1.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (3.75 - 6.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.53 - 7.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.98 - 6.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.732242079500610378975475837896, −9.160952969149824103357753927442, −8.352556735808064018307536479660, −7.33147156121341853432986716143, −7.08823171045706843602799524133, −5.37167341409993802679617325332, −4.60830114571579076370437984628, −3.66843940760103763017763795582, −2.52240966077642653411811289599, −1.34545836786369813739461595908, 1.48848888189180257337448790869, 2.43227705628610174409541359121, 3.70029420914446370702396553005, 4.46035095721406054117412520271, 5.83316071279499733105020892916, 6.53486574665021697497328208044, 7.933739112095196500474965003037, 8.226013760569711312216291453203, 8.847681484256276632176361907720, 10.11150057872520475571561324303

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