↑

Properties

Label 2-1008-63.4-c1-0-8
Degree $2$
Conductor $1008$
Sign $0.999 + 0.0184i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.52 − 0.812i)3-s − 3.79·5-s + (−2.59 − 0.525i)7-s + (1.67 + 2.48i)9-s − 4.51·11-s + (0.588 − 1.01i)13-s + (5.81 + 3.08i)15-s + (−2.95 + 5.12i)17-s + (−2.55 − 4.42i)19-s + (3.53 + 2.91i)21-s + 4.18·23-s + 9.43·25-s + (−0.545 − 5.16i)27-s + (2.11 + 3.65i)29-s + (−3.12 − 5.40i)31-s + ⋯
L(s)  = 1  + (−0.883 − 0.469i)3-s − 1.69·5-s + (−0.980 − 0.198i)7-s + (0.559 + 0.828i)9-s − 1.36·11-s + (0.163 − 0.282i)13-s + (1.50 + 0.797i)15-s + (−0.717 + 1.24i)17-s + (−0.586 − 1.01i)19-s + (0.772 + 0.635i)21-s + 0.871·23-s + 1.88·25-s + (−0.105 − 0.994i)27-s + (0.392 + 0.679i)29-s + (−0.560 − 0.971i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.999 + 0.0184i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.999 + 0.0184i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3429632061\)
\(L(\frac12)\) \(\approx\) \(0.3429632061\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.52 + 0.812i)T \)
7 \( 1 + (2.59 + 0.525i)T \)
good5 \( 1 + 3.79T + 5T^{2} \)
11 \( 1 + 4.51T + 11T^{2} \)
13 \( 1 + (-0.588 + 1.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.95 - 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.55 + 4.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.18T + 23T^{2} \)
29 \( 1 + (-2.11 - 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.12 + 5.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.754 + 1.30i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.01 - 8.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.11 - 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.49 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.19 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.729 - 1.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.813 - 1.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.920 - 1.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.307 - 0.532i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.25 + 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 - 4.10i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.41996651971926025266618112274, −8.952560480830351903165058156812, −8.107569524102095709927516344095, −7.33855018768937896649343581984, −6.76645848564077418967102925196, −5.71321649902192129827199780332, −4.65856766383592166298394619047, −3.81455585671978767960108932059, −2.60757872659080282038203842557, −0.51806063536615602758493698416, 0.38802610501954551985274282081, 2.88316864178519326091714348396, 3.80218347251144303215288725158, 4.67511289961983563453342209752, 5.52222352339601985501097984407, 6.73718986832645100782924262420, 7.25978632717203265956090936206, 8.326664709396432935807472779719, 9.142672028742474885570088801085, 10.21784571947437323984404429547

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