↑

Properties

Label 2-2900-29.28-c1-0-26
Degree $2$
Conductor $2900$
Sign $0.602 + 0.798i$
Analytic cond. $23.1566$
Root an. cond. $4.81213$
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  + 1.78i·3-s − 3.41·7-s − 0.171·9-s + 4.29i·11-s − 5.65·13-s − 0.737i·17-s − 6.81i·19-s − 6.08i·21-s + 2.58·23-s + 5.03i·27-s + (−3.24 − 4.29i)29-s − 1.78i·31-s − 7.65·33-s − 4.29i·37-s − 10.0i·39-s + ⋯
L(s)  = 1  + 1.02i·3-s − 1.29·7-s − 0.0571·9-s + 1.29i·11-s − 1.56·13-s − 0.178i·17-s − 1.56i·19-s − 1.32i·21-s + 0.539·23-s + 0.969i·27-s + (−0.602 − 0.798i)29-s − 0.319i·31-s − 1.33·33-s − 0.706i·37-s − 1.61i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.602 + 0.798i$
Analytic conductor: \(23.1566\)
Root analytic conductor: \(4.81213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2900,\ (\ :1/2),\ 0.602 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6180600535\)
\(L(\frac12)\) \(\approx\) \(0.6180600535\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 + (3.24 + 4.29i)T \)
good3 \( 1 - 1.78iT - 3T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 - 4.29iT - 11T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + 0.737iT - 17T^{2} \)
19 \( 1 + 6.81iT - 19T^{2} \)
23 \( 1 - 2.58T + 23T^{2} \)
31 \( 1 + 1.78iT - 31T^{2} \)
37 \( 1 + 4.29iT - 37T^{2} \)
41 \( 1 - 6.08iT - 41T^{2} \)
43 \( 1 - 1.78iT - 43T^{2} \)
47 \( 1 + 1.78iT - 47T^{2} \)
53 \( 1 + 1.17T + 53T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + 6.08iT - 61T^{2} \)
67 \( 1 - 9.41T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + 2.92T + 83T^{2} \)
89 \( 1 - 10.0iT - 89T^{2} \)
97 \( 1 + 11.8iT - 97T^{2} \)
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

−9.175068634008592878187395203894, −7.74971385734610906952812723499, −7.06308193537553076539470041516, −6.56428629547910755776783615228, −5.25930932078946366305063563785, −4.74453231226977904367084569338, −4.01854114413605941126147948687, −2.99844922356877821457016338610, −2.22347691548257605121659829088, −0.22199308890375243577072616096, 1.00509327108784632863181990008, 2.23721932595363770990074025546, 3.15164588201126996283750214064, 3.92119742200597450978131261164, 5.28820644813168305266588769462, 5.94201746360478940412481837891, 6.76612121625215768013698154257, 7.19411463867724090626036116187, 8.050751951936911894013683015833, 8.738798428093295818448731729077

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