↑

Properties

Label 2-40e2-80.27-c1-0-33
Degree $2$
Conductor $1600$
Sign $-0.997 - 0.0680i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
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  + 2.02·3-s + (−3.26 − 3.26i)7-s + 1.11·9-s + (−3.55 + 3.55i)11-s + 1.41i·13-s + (−1.73 − 1.73i)17-s + (−4.19 + 4.19i)19-s + (−6.63 − 6.63i)21-s + (0.177 − 0.177i)23-s − 3.82·27-s + (−1.63 − 1.63i)29-s − 8.15i·31-s + (−7.20 + 7.20i)33-s + 1.34i·37-s + 2.87i·39-s + ⋯
L(s)  = 1  + 1.17·3-s + (−1.23 − 1.23i)7-s + 0.371·9-s + (−1.07 + 1.07i)11-s + 0.393i·13-s + (−0.420 − 0.420i)17-s + (−0.962 + 0.962i)19-s + (−1.44 − 1.44i)21-s + (0.0370 − 0.0370i)23-s − 0.735·27-s + (−0.304 − 0.304i)29-s − 1.46i·31-s + (−1.25 + 1.25i)33-s + 0.220i·37-s + 0.460i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.997 - 0.0680i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.997 - 0.0680i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1134379205\)
\(L(\frac12)\) \(\approx\) \(0.1134379205\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 2.02T + 3T^{2} \)
7 \( 1 + (3.26 + 3.26i)T + 7iT^{2} \)
11 \( 1 + (3.55 - 3.55i)T - 11iT^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
17 \( 1 + (1.73 + 1.73i)T + 17iT^{2} \)
19 \( 1 + (4.19 - 4.19i)T - 19iT^{2} \)
23 \( 1 + (-0.177 + 0.177i)T - 23iT^{2} \)
29 \( 1 + (1.63 + 1.63i)T + 29iT^{2} \)
31 \( 1 + 8.15iT - 31T^{2} \)
37 \( 1 - 1.34iT - 37T^{2} \)
41 \( 1 + 1.16iT - 41T^{2} \)
43 \( 1 + 1.04iT - 43T^{2} \)
47 \( 1 + (-4.25 + 4.25i)T - 47iT^{2} \)
53 \( 1 + 8.19T + 53T^{2} \)
59 \( 1 + (3.96 + 3.96i)T + 59iT^{2} \)
61 \( 1 + (7.22 - 7.22i)T - 61iT^{2} \)
67 \( 1 - 9.19iT - 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-5.86 - 5.86i)T + 73iT^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + (5.71 + 5.71i)T + 97iT^{2} \)
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

−9.125361871852526120293591921902, −8.083044728468994314050504212100, −7.51189839142361841658648307236, −6.82765714138063551140758320050, −5.86197740635213314964855238317, −4.43969739712712914471678235036, −3.86138951693097755840630984956, −2.85621363403512870778985207190, −2.00793390602199500522158399231, −0.03293174547556144992205805705, 2.19759114901802290082862911435, 2.97289036388981205388558364805, 3.38914914350159467057811227104, 4.89474175985750446858017500085, 5.88026099352451721257436805661, 6.48252481780088600156948535602, 7.67276361900131905971297078362, 8.468132922065411207084815711423, 8.925527159311924307152530882003, 9.460446548519099660708838276693

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