↑

Properties

Label 2-20e2-80.3-c1-0-21
Degree $2$
Conductor $400$
Sign $0.250 + 0.968i$
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  + (1.38 − 0.307i)2-s − 2.85·3-s + (1.81 − 0.849i)4-s + (−3.94 + 0.879i)6-s + (0.458 − 0.458i)7-s + (2.23 − 1.73i)8-s + 5.15·9-s + (−0.492 − 0.492i)11-s + (−5.17 + 2.42i)12-s − 4.52i·13-s + (0.492 − 0.774i)14-s + (2.55 − 3.07i)16-s + (3.12 − 3.12i)17-s + (7.11 − 1.58i)18-s + (−4.04 − 4.04i)19-s + ⋯
L(s)  = 1  + (0.976 − 0.217i)2-s − 1.64·3-s + (0.905 − 0.424i)4-s + (−1.60 + 0.358i)6-s + (0.173 − 0.173i)7-s + (0.791 − 0.611i)8-s + 1.71·9-s + (−0.148 − 0.148i)11-s + (−1.49 + 0.700i)12-s − 1.25i·13-s + (0.131 − 0.207i)14-s + (0.638 − 0.769i)16-s + (0.758 − 0.758i)17-s + (1.67 − 0.374i)18-s + (−0.928 − 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.250 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16196 - 0.899630i\)
\(L(\frac12)\) \(\approx\) \(1.16196 - 0.899630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.307i)T \)
5 \( 1 \)
good3 \( 1 + 2.85T + 3T^{2} \)
7 \( 1 + (-0.458 + 0.458i)T - 7iT^{2} \)
11 \( 1 + (0.492 + 0.492i)T + 11iT^{2} \)
13 \( 1 + 4.52iT - 13T^{2} \)
17 \( 1 + (-3.12 + 3.12i)T - 17iT^{2} \)
19 \( 1 + (4.04 + 4.04i)T + 19iT^{2} \)
23 \( 1 + (-1.80 - 1.80i)T + 23iT^{2} \)
29 \( 1 + (-3.83 + 3.83i)T - 29iT^{2} \)
31 \( 1 - 0.139iT - 31T^{2} \)
37 \( 1 - 5.84iT - 37T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 - 7.49iT - 43T^{2} \)
47 \( 1 + (4.14 + 4.14i)T + 47iT^{2} \)
53 \( 1 + 2.75T + 53T^{2} \)
59 \( 1 + (-3.62 + 3.62i)T - 59iT^{2} \)
61 \( 1 + (-3.72 - 3.72i)T + 61iT^{2} \)
67 \( 1 - 3.32iT - 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (2.55 - 2.55i)T - 73iT^{2} \)
79 \( 1 - 3.86T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + (-4.95 + 4.95i)T - 97iT^{2} \)
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

−11.31953987308109066620549489689, −10.53164965207885290622539179364, −9.856238363794824186656465634503, −7.951247494856675735841394392395, −6.87928539751904099627160474578, −6.05977145229149304861792362019, −5.20151749849600853232360143516, −4.54028723283179908312881725981, −2.96066014771411380492948112549, −0.920462533132947049791136011054, 1.81573607910301803033505251590, 3.87082927231339987849934388910, 4.80076011881956921889833718164, 5.69664680381388498789793894689, 6.43745935882619875149888685182, 7.22165822941366930157026126074, 8.532203240616232816001188703944, 10.17952344464371833466827730582, 10.83271368390021916920667595726, 11.63878538473181743266079619641

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