Properties

Label 2-2e9-512.157-c1-0-36
Degree $2$
Conductor $512$
Sign $0.300 - 0.953i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
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  + (1.15 + 0.810i)2-s + (2.35 − 0.906i)3-s + (0.684 + 1.87i)4-s + (−2.18 + 2.79i)5-s + (3.45 + 0.855i)6-s + (−0.0220 + 0.0616i)7-s + (−0.730 + 2.73i)8-s + (2.48 − 2.25i)9-s + (−4.79 + 1.47i)10-s + (−0.212 − 0.948i)11-s + (3.31 + 3.79i)12-s + (0.277 − 0.0767i)13-s + (−0.0755 + 0.0535i)14-s + (−2.59 + 8.55i)15-s + (−3.06 + 2.57i)16-s + (6.31 − 1.91i)17-s + ⋯
L(s)  = 1  + (0.819 + 0.573i)2-s + (1.35 − 0.523i)3-s + (0.342 + 0.939i)4-s + (−0.976 + 1.25i)5-s + (1.41 + 0.349i)6-s + (−0.00833 + 0.0233i)7-s + (−0.258 + 0.966i)8-s + (0.827 − 0.750i)9-s + (−1.51 + 0.465i)10-s + (−0.0642 − 0.285i)11-s + (0.956 + 1.09i)12-s + (0.0768 − 0.0212i)13-s + (−0.0201 + 0.0143i)14-s + (−0.670 + 2.21i)15-s + (−0.765 + 0.643i)16-s + (1.53 − 0.464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.36532 + 1.73556i\)
\(L(\frac12)\) \(\approx\) \(2.36532 + 1.73556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.810i)T \)
good3 \( 1 + (-2.35 + 0.906i)T + (2.22 - 2.01i)T^{2} \)
5 \( 1 + (2.18 - 2.79i)T + (-1.21 - 4.85i)T^{2} \)
7 \( 1 + (0.0220 - 0.0616i)T + (-5.41 - 4.44i)T^{2} \)
11 \( 1 + (0.212 + 0.948i)T + (-9.94 + 4.70i)T^{2} \)
13 \( 1 + (-0.277 + 0.0767i)T + (11.1 - 6.68i)T^{2} \)
17 \( 1 + (-6.31 + 1.91i)T + (14.1 - 9.44i)T^{2} \)
19 \( 1 + (0.163 + 0.325i)T + (-11.3 + 15.2i)T^{2} \)
23 \( 1 + (-1.56 + 0.231i)T + (22.0 - 6.67i)T^{2} \)
29 \( 1 + (1.49 + 8.59i)T + (-27.3 + 9.76i)T^{2} \)
31 \( 1 + (2.16 - 3.24i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.25 - 1.08i)T + (5.42 - 36.5i)T^{2} \)
41 \( 1 + (-1.88 + 7.51i)T + (-36.1 - 19.3i)T^{2} \)
43 \( 1 + (2.39 + 5.40i)T + (-28.8 + 31.8i)T^{2} \)
47 \( 1 + (-0.711 - 0.867i)T + (-9.16 + 46.0i)T^{2} \)
53 \( 1 + (-1.15 + 6.65i)T + (-49.9 - 17.8i)T^{2} \)
59 \( 1 + (-1.89 + 6.86i)T + (-50.6 - 30.3i)T^{2} \)
61 \( 1 + (-6.89 - 0.169i)T + (60.9 + 2.99i)T^{2} \)
67 \( 1 + (9.87 - 9.40i)T + (3.28 - 66.9i)T^{2} \)
71 \( 1 + (12.2 - 0.599i)T + (70.6 - 6.95i)T^{2} \)
73 \( 1 + (-2.20 - 6.16i)T + (-56.4 + 46.3i)T^{2} \)
79 \( 1 + (-14.1 + 1.39i)T + (77.4 - 15.4i)T^{2} \)
83 \( 1 + (5.57 + 4.80i)T + (12.1 + 82.1i)T^{2} \)
89 \( 1 + (14.5 + 2.16i)T + (85.1 + 25.8i)T^{2} \)
97 \( 1 + (2.31 - 0.460i)T + (89.6 - 37.1i)T^{2} \)
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

−11.36739783689869413594879966532, −10.23650660114642515206739741366, −8.877582236891229759797849980297, −7.990686372697995211702110843016, −7.46588726495347627226548707375, −6.84560408567466195224462350314, −5.57797021692853315872704958144, −3.92597183048482705524690242508, −3.28280063184653565497082499211, −2.47033893072392596526575478762, 1.40833520321466367251136372699, 3.05325257393999504781321057181, 3.83641989008176508647981138809, 4.61826029528554534104338327638, 5.60687533823500214438945827478, 7.34182927106969545390787744792, 8.186010814478266746372188192720, 9.036227145367264773948761170863, 9.738459770871375047206750840506, 10.71828827312739678069778034605

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