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Properties

Label 2-20e2-80.27-c1-0-1
Degree $2$
Conductor $400$
Sign $-0.852 + 0.522i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.516 + 1.31i)2-s − 1.28·3-s + (−1.46 + 1.36i)4-s + (−0.662 − 1.68i)6-s + (1.13 + 1.13i)7-s + (−2.54 − 1.22i)8-s − 1.35·9-s + (−2.32 + 2.32i)11-s + (1.87 − 1.74i)12-s + 1.36i·13-s + (−0.911 + 2.08i)14-s + (0.297 − 3.98i)16-s + (−5.25 − 5.25i)17-s + (−0.702 − 1.78i)18-s + (−3.69 + 3.69i)19-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s − 0.739·3-s + (−0.732 + 0.680i)4-s + (−0.270 − 0.688i)6-s + (0.430 + 0.430i)7-s + (−0.901 − 0.433i)8-s − 0.452·9-s + (−0.700 + 0.700i)11-s + (0.542 − 0.503i)12-s + 0.378i·13-s + (−0.243 + 0.558i)14-s + (0.0744 − 0.997i)16-s + (−1.27 − 1.27i)17-s + (−0.165 − 0.421i)18-s + (−0.848 + 0.848i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.852 + 0.522i$
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: \((2,\ 400,\ (\ :1/2),\ -0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134445 - 0.476462i\)
\(L(\frac12)\) \(\approx\) \(0.134445 - 0.476462i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.516 - 1.31i)T \)
5 \( 1 \)
good3 \( 1 + 1.28T + 3T^{2} \)
7 \( 1 + (-1.13 - 1.13i)T + 7iT^{2} \)
11 \( 1 + (2.32 - 2.32i)T - 11iT^{2} \)
13 \( 1 - 1.36iT - 13T^{2} \)
17 \( 1 + (5.25 + 5.25i)T + 17iT^{2} \)
19 \( 1 + (3.69 - 3.69i)T - 19iT^{2} \)
23 \( 1 + (-0.911 + 0.911i)T - 23iT^{2} \)
29 \( 1 + (2.37 + 2.37i)T + 29iT^{2} \)
31 \( 1 - 0.242iT - 31T^{2} \)
37 \( 1 - 3.34iT - 37T^{2} \)
41 \( 1 - 2.66iT - 41T^{2} \)
43 \( 1 - 9.04iT - 43T^{2} \)
47 \( 1 + (7.87 - 7.87i)T - 47iT^{2} \)
53 \( 1 - 5.80T + 53T^{2} \)
59 \( 1 + (-5.91 - 5.91i)T + 59iT^{2} \)
61 \( 1 + (6.67 - 6.67i)T - 61iT^{2} \)
67 \( 1 - 4.54iT - 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + (-1.49 - 1.49i)T + 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 3.26T + 83T^{2} \)
89 \( 1 - 9.77T + 89T^{2} \)
97 \( 1 + (-1.63 - 1.63i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.81630498605627132285577783573, −11.17444698118621868883211059043, −9.876224906570016057325113423935, −8.842515301124894235289133681527, −8.017469817010241876652766460785, −6.92397918348966382487487524857, −6.09244807880241580698983125205, −5.10054280661950354765182904822, −4.42718921705749732584116080797, −2.60459308473230475507807198640, 0.29832917399720241200395437541, 2.19029148192097493019992065440, 3.63297784630383949777833783905, 4.82273362541843107169420061826, 5.64492723643771937971862297712, 6.63544444743841585455299456439, 8.273465309726422503399053778723, 8.928604144119519205837713489170, 10.40373713068341588294277886030, 10.91753619859117369823571284662

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