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Properties

Label 2-20e2-25.14-c1-0-10
Degree $2$
Conductor $400$
Sign $-0.642 + 0.766i$
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  + (−1.92 − 0.626i)3-s + (1.20 − 1.88i)5-s + 0.498i·7-s + (0.899 + 0.653i)9-s + (1.49 − 1.08i)11-s + (0.541 − 0.745i)13-s + (−3.50 + 2.87i)15-s + (−6.51 + 2.11i)17-s + (−1.86 − 5.72i)19-s + (0.312 − 0.960i)21-s + (−4.56 − 6.28i)23-s + (−2.09 − 4.53i)25-s + (2.25 + 3.09i)27-s + (1.30 − 4.02i)29-s + (0.963 + 2.96i)31-s + ⋯
L(s)  = 1  + (−1.11 − 0.361i)3-s + (0.538 − 0.842i)5-s + 0.188i·7-s + (0.299 + 0.217i)9-s + (0.449 − 0.326i)11-s + (0.150 − 0.206i)13-s + (−0.904 + 0.743i)15-s + (−1.58 + 0.513i)17-s + (−0.427 − 1.31i)19-s + (0.0681 − 0.209i)21-s + (−0.951 − 1.30i)23-s + (−0.419 − 0.907i)25-s + (0.433 + 0.596i)27-s + (0.243 − 0.748i)29-s + (0.173 + 0.532i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307074 - 0.658146i\)
\(L(\frac12)\) \(\approx\) \(0.307074 - 0.658146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.88i)T \)
good3 \( 1 + (1.92 + 0.626i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 0.498iT - 7T^{2} \)
11 \( 1 + (-1.49 + 1.08i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.541 + 0.745i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (6.51 - 2.11i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.86 + 5.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.56 + 6.28i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.30 + 4.02i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.963 - 2.96i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.83 - 2.52i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.94 + 2.86i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.22iT - 43T^{2} \)
47 \( 1 + (-2.19 - 0.712i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.8 + 3.52i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.58 - 4.06i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.42 + 3.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.761 + 0.247i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.83 - 5.64i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.49 - 11.6i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.60 - 11.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-13.3 + 4.33i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-14.2 + 10.3i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.40 - 2.08i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.06795162094198470385208583100, −10.22392486711935309724225897716, −8.918373078574396987646330246045, −8.470030145648641010042733456858, −6.71067274300287002947612781806, −6.25682916954719553977737626981, −5.20545422682181873632093402966, −4.27919793742178833616160418096, −2.20564736210077091218825327682, −0.52811169773932561922037339044, 2.00317873142487603250376363721, 3.72567670396469544696261833877, 4.89004558777595934213345366410, 6.04054363644786265475606277245, 6.55762156724420115203268501377, 7.70714000676561584605326773878, 9.139667616629552026807774952987, 10.02987549590352264823312840364, 10.75415481398119793037417539245, 11.43525951575898749042715104770

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