↑

Properties

Label 2-1008-63.59-c1-0-24
Degree $2$
Conductor $1008$
Sign $0.956 + 0.291i$
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  + (−1.24 + 1.20i)3-s + 2.04·5-s + (−1.41 + 2.23i)7-s + (0.102 − 2.99i)9-s − 5.90i·11-s + (−0.139 − 0.0804i)13-s + (−2.55 + 2.46i)15-s + (2.77 − 4.81i)17-s + (4.02 − 2.32i)19-s + (−0.932 − 4.48i)21-s − 0.433i·23-s − 0.801·25-s + (3.48 + 3.85i)27-s + (−1.95 + 1.12i)29-s + (2.57 − 1.48i)31-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)3-s + 0.916·5-s + (−0.534 + 0.845i)7-s + (0.0341 − 0.999i)9-s − 1.78i·11-s + (−0.0386 − 0.0223i)13-s + (−0.658 + 0.636i)15-s + (0.674 − 1.16i)17-s + (0.922 − 0.532i)19-s + (−0.203 − 0.979i)21-s − 0.0903i·23-s − 0.160·25-s + (0.669 + 0.742i)27-s + (−0.363 + 0.209i)29-s + (0.463 − 0.267i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.291i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.956 + 0.291i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.956 + 0.291i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.324111876\)
\(L(\frac12)\) \(\approx\) \(1.324111876\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 - 1.20i)T \)
7 \( 1 + (1.41 - 2.23i)T \)
good5 \( 1 - 2.04T + 5T^{2} \)
11 \( 1 + 5.90iT - 11T^{2} \)
13 \( 1 + (0.139 + 0.0804i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.77 + 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.02 + 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.433iT - 23T^{2} \)
29 \( 1 + (1.95 - 1.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.57 + 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 3.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.35 + 4.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0650 - 0.112i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-10.7 - 6.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 - 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.98 + 3.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.64 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.48iT - 71T^{2} \)
73 \( 1 + (2.60 + 1.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.69 + 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.62 + 13.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.04 - 7.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.61 + 1.50i)T + (48.5 - 84.0i)T^{2} \)
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

−9.866376474546178683025328639451, −9.246360197468040516505784642722, −8.640865388992377433291559034043, −7.22862430205111984800514043327, −6.05760880266294370976266201660, −5.75021415437612313497042235747, −4.99923466485868116260969745920, −3.49096723467003587006320911175, −2.71352348304417389596214519620, −0.75100083530228814737004643276, 1.27807786364267592713689625793, 2.21235994566358934347419155530, 3.82979547549095157912896307083, 4.94799536652708541156275590121, 5.84698767285463686130380060903, 6.59952883983131863575267964412, 7.37060978991974498822763801097, 8.025430521245510736027799813405, 9.639201668673246012272157335262, 9.929704608807586586355219383098

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