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Properties

Label 2-20e2-25.4-c1-0-12
Degree $2$
Conductor $400$
Sign $-0.158 + 0.987i$
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  + (0.865 − 1.19i)3-s + (0.107 − 2.23i)5-s − 3.26i·7-s + (0.257 + 0.792i)9-s + (−0.618 + 1.90i)11-s + (0.281 − 0.0915i)13-s + (−2.56 − 2.06i)15-s + (−3.03 − 4.17i)17-s + (−1.39 + 1.01i)19-s + (−3.88 − 2.82i)21-s + (−0.836 − 0.271i)23-s + (−4.97 − 0.480i)25-s + (5.36 + 1.74i)27-s + (4.78 + 3.47i)29-s + (4.93 − 3.58i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.687i)3-s + (0.0481 − 0.998i)5-s − 1.23i·7-s + (0.0858 + 0.264i)9-s + (−0.186 + 0.573i)11-s + (0.0781 − 0.0254i)13-s + (−0.662 − 0.532i)15-s + (−0.736 − 1.01i)17-s + (−0.321 + 0.233i)19-s + (−0.847 − 0.615i)21-s + (−0.174 − 0.0566i)23-s + (−0.995 − 0.0961i)25-s + (1.03 + 0.335i)27-s + (0.888 + 0.645i)29-s + (0.886 − 0.643i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.158 + 0.987i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998212 - 1.17123i\)
\(L(\frac12)\) \(\approx\) \(0.998212 - 1.17123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.107 + 2.23i)T \)
good3 \( 1 + (-0.865 + 1.19i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + 3.26iT - 7T^{2} \)
11 \( 1 + (0.618 - 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.281 + 0.0915i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.03 + 4.17i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.39 - 1.01i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.836 + 0.271i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.78 - 3.47i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.93 + 3.58i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-7.69 + 2.49i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.313 + 0.965i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.24iT - 43T^{2} \)
47 \( 1 + (-2.48 + 3.41i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.76 - 6.55i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.83 - 5.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.282 + 0.870i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (4.04 + 5.57i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.82 + 3.50i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.40 - 2.72i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.27 - 4.56i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.53 - 11.7i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.32 - 7.15i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.95 + 5.44i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.98639596835533635908645428465, −10.05952962022723824269053081915, −9.114307545646413710902688594789, −8.070336050252921342700639922860, −7.47801512902496910337510395366, −6.50748733253458946906143492383, −4.93188781254292799949316759933, −4.19791805314048081565769626235, −2.43137588547821310809104578555, −1.02018607115008386097177684968, 2.39666971566014814266682902032, 3.29774250502416762595333238906, 4.48818165847184404110438957205, 5.98153072745159796697553782979, 6.57234376127009895412062556986, 8.132750480335867381609657836471, 8.784950491751788594536566272487, 9.733520212492472046533391560117, 10.52085631300779256374467935364, 11.41038015850413039042016688794

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