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Properties

Label 2-1008-63.58-c1-0-25
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: $\chi_{1008} (625, \cdot )$
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

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

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