Properties

Label 2-20e2-400.227-c1-0-31
Degree $2$
Conductor $400$
Sign $0.727 + 0.685i$
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  + (1.32 − 0.499i)2-s + (−2.67 − 0.867i)3-s + (1.50 − 1.32i)4-s + (2.16 + 0.573i)5-s + (−3.96 + 0.185i)6-s + (3.07 + 3.07i)7-s + (1.32 − 2.49i)8-s + (3.95 + 2.87i)9-s + (3.14 − 0.320i)10-s + (−2.14 − 0.340i)11-s + (−5.15 + 2.22i)12-s + (0.756 + 0.549i)13-s + (5.60 + 2.53i)14-s + (−5.27 − 3.40i)15-s + (0.505 − 3.96i)16-s + (2.82 − 5.53i)17-s + ⋯
L(s)  = 1  + (0.935 − 0.353i)2-s + (−1.54 − 0.501i)3-s + (0.750 − 0.660i)4-s + (0.966 + 0.256i)5-s + (−1.61 + 0.0759i)6-s + (1.16 + 1.16i)7-s + (0.468 − 0.883i)8-s + (1.31 + 0.957i)9-s + (0.994 − 0.101i)10-s + (−0.647 − 0.102i)11-s + (−1.48 + 0.643i)12-s + (0.209 + 0.152i)13-s + (1.49 + 0.677i)14-s + (−1.36 − 0.880i)15-s + (0.126 − 0.991i)16-s + (0.684 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.727 + 0.685i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.727 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77940 - 0.706500i\)
\(L(\frac12)\) \(\approx\) \(1.77940 - 0.706500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.499i)T \)
5 \( 1 + (-2.16 - 0.573i)T \)
good3 \( 1 + (2.67 + 0.867i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-3.07 - 3.07i)T + 7iT^{2} \)
11 \( 1 + (2.14 + 0.340i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (-0.756 - 0.549i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.82 + 5.53i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.30 - 4.53i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-1.10 + 6.94i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.330 + 0.647i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (-1.32 + 0.429i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.950 + 0.690i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.35 - 8.74i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + (-0.702 - 1.37i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-2.64 - 0.858i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.39 + 1.01i)T + (56.1 - 18.2i)T^{2} \)
61 \( 1 + (6.44 + 1.02i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (-0.0812 - 0.249i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.01 - 6.20i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.36 - 1.32i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.800 - 2.46i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-8.32 + 2.70i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (13.5 - 9.85i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (5.19 - 2.64i)T + (57.0 - 78.4i)T^{2} \)
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

−11.40040708911730763048594939785, −10.64401649187920026934848339027, −9.843150308618298848001871843550, −8.238313107665494934681711897497, −6.85728293352557133465045549624, −6.07699999697988573883110046808, −5.29754845882155685533222805362, −4.87301581737270932179996348176, −2.61928954042569740606405514980, −1.49798225504691032959389215293, 1.58045032495796921486143867047, 3.81946735920495652052007864924, 4.96907423854983692637158746147, 5.29471975405639596841802118323, 6.32977518358436786725211946293, 7.26600529742039790975527971985, 8.415000107735152110652266693107, 10.17104681054381927579339222201, 10.64067296575530655093572450810, 11.32403468315222384440081732463

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