↑

Properties

Label 2-20e2-80.27-c1-0-30
Degree $2$
Conductor $400$
Sign $-0.466 + 0.884i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
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  + (1.23 − 0.687i)2-s − 0.614·3-s + (1.05 − 1.69i)4-s + (−0.759 + 0.422i)6-s + (−2.83 − 2.83i)7-s + (0.134 − 2.82i)8-s − 2.62·9-s + (1.95 − 1.95i)11-s + (−0.647 + 1.04i)12-s − 2.05i·13-s + (−5.45 − 1.55i)14-s + (−1.77 − 3.58i)16-s + (4.06 + 4.06i)17-s + (−3.24 + 1.80i)18-s + (−0.683 + 0.683i)19-s + ⋯
L(s)  = 1  + (0.873 − 0.486i)2-s − 0.354·3-s + (0.527 − 0.849i)4-s + (−0.310 + 0.172i)6-s + (−1.07 − 1.07i)7-s + (0.0473 − 0.998i)8-s − 0.874·9-s + (0.590 − 0.590i)11-s + (−0.187 + 0.301i)12-s − 0.569i·13-s + (−1.45 − 0.415i)14-s + (−0.444 − 0.895i)16-s + (0.986 + 0.986i)17-s + (−0.763 + 0.425i)18-s + (−0.156 + 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.466 + 0.884i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851079 - 1.41193i\)
\(L(\frac12)\) \(\approx\) \(0.851079 - 1.41193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.687i)T \)
5 \( 1 \)
good3 \( 1 + 0.614T + 3T^{2} \)
7 \( 1 + (2.83 + 2.83i)T + 7iT^{2} \)
11 \( 1 + (-1.95 + 1.95i)T - 11iT^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (0.683 - 0.683i)T - 19iT^{2} \)
23 \( 1 + (-4.95 + 4.95i)T - 23iT^{2} \)
29 \( 1 + (-0.835 - 0.835i)T + 29iT^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 0.849iT - 43T^{2} \)
47 \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \)
53 \( 1 + 5.17T + 53T^{2} \)
59 \( 1 + (4.16 + 4.16i)T + 59iT^{2} \)
61 \( 1 + (-5.55 + 5.55i)T - 61iT^{2} \)
67 \( 1 - 1.73iT - 67T^{2} \)
71 \( 1 - 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 2.75T + 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (-3.52 - 3.52i)T + 97iT^{2} \)
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

−10.86828689504005430168776353915, −10.49070217489898462309830846272, −9.473760180766338313839346177604, −8.165108348715465357982130375238, −6.70420253325302579505043977655, −6.19896068843508944147079214321, −5.09502812078113156810200776542, −3.73005821425873487837261832469, −3.04169992658191721091128640352, −0.870736805512496636914497877479, 2.53447887766315144566709971411, 3.53030121150387709526144052725, 5.01985253558891569829477485969, 5.80258689488522695831034028875, 6.60645621607651629040931800284, 7.54979449721965069560681643100, 8.942521648761133605282110011926, 9.479643885741613512625922937133, 11.04559554846205370066932065970, 11.96078450861194275983484430024

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