↑

Properties

Label 2-1008-252.191-c1-0-33
Degree $2$
Conductor $1008$
Sign $0.768 + 0.639i$
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  + (1.51 − 0.832i)3-s + 1.27i·5-s + (2.64 + 0.0416i)7-s + (1.61 − 2.52i)9-s − 1.83·11-s + (−3.03 − 5.26i)13-s + (1.05 + 1.93i)15-s + (4.50 − 2.60i)17-s + (−3.63 − 2.10i)19-s + (4.05 − 2.14i)21-s + 4.61·23-s + 3.38·25-s + (0.341 − 5.18i)27-s + (7.74 + 4.46i)29-s + (0.317 + 0.183i)31-s + ⋯
L(s)  = 1  + (0.876 − 0.480i)3-s + 0.568i·5-s + (0.999 + 0.0157i)7-s + (0.537 − 0.843i)9-s − 0.552·11-s + (−0.842 − 1.45i)13-s + (0.273 + 0.498i)15-s + (1.09 − 0.631i)17-s + (−0.834 − 0.481i)19-s + (0.884 − 0.467i)21-s + 0.961·23-s + 0.676·25-s + (0.0657 − 0.997i)27-s + (1.43 + 0.829i)29-s + (0.0569 + 0.0328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.768 + 0.639i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.768 + 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.356915826\)
\(L(\frac12)\) \(\approx\) \(2.356915826\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.51 + 0.832i)T \)
7 \( 1 + (-2.64 - 0.0416i)T \)
good5 \( 1 - 1.27iT - 5T^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 + (3.03 + 5.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.50 + 2.60i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.63 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.61T + 23T^{2} \)
29 \( 1 + (-7.74 - 4.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.317 - 0.183i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.03 - 6.99i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.37 - 2.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.09 + 2.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.83 - 8.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.2 + 5.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.08 + 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.70 - 6.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.23 + 2.44i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-2.13 - 3.69i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.1 - 5.85i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.73 - 6.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.5 + 6.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.34 + 2.33i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.12230547914649277564873005674, −8.711826447245449423613691486770, −8.263749976953193254812599403281, −7.36405524879139717905113012656, −6.87533833063570063700947885660, −5.42720698538477759825330528930, −4.68261411989943002324925683708, −3.06979710916144516642324077555, −2.68609236242837179687303592838, −1.11455519393870742449981116871, 1.57706345336634864642978283129, 2.56927898645677378667656890053, 3.98790695819878274224736556130, 4.68197507437462402855175178013, 5.42505634680132224258334945094, 6.92996413576284892361981826283, 7.74651495308568458727361713777, 8.588798594365547011748353909374, 8.955548299577755280765685589309, 10.14729817801420431379788930061

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