Properties

Label 2-20e2-25.14-c1-0-2
Degree $2$
Conductor $400$
Sign $-0.635 - 0.772i$
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  + (0.685 + 0.222i)3-s + (−2.21 + 0.277i)5-s + 4.42i·7-s + (−2.00 − 1.45i)9-s + (−3.36 + 2.44i)11-s + (0.592 − 0.815i)13-s + (−1.58 − 0.304i)15-s + (−3.49 + 1.13i)17-s + (−0.865 − 2.66i)19-s + (−0.985 + 3.03i)21-s + (4.41 + 6.08i)23-s + (4.84 − 1.22i)25-s + (−2.32 − 3.19i)27-s + (−2.81 + 8.66i)29-s + (0.593 + 1.82i)31-s + ⋯
L(s)  = 1  + (0.395 + 0.128i)3-s + (−0.992 + 0.123i)5-s + 1.67i·7-s + (−0.668 − 0.485i)9-s + (−1.01 + 0.737i)11-s + (0.164 − 0.226i)13-s + (−0.408 − 0.0785i)15-s + (−0.848 + 0.275i)17-s + (−0.198 − 0.610i)19-s + (−0.215 + 0.661i)21-s + (0.921 + 1.26i)23-s + (0.969 − 0.245i)25-s + (−0.446 − 0.615i)27-s + (−0.522 + 1.60i)29-s + (0.106 + 0.328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.635 - 0.772i$
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: \((2,\ 400,\ (\ :1/2),\ -0.635 - 0.772i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339183 + 0.718185i\)
\(L(\frac12)\) \(\approx\) \(0.339183 + 0.718185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.21 - 0.277i)T \)
good3 \( 1 + (-0.685 - 0.222i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 4.42iT - 7T^{2} \)
11 \( 1 + (3.36 - 2.44i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.592 + 0.815i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.49 - 1.13i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.865 + 2.66i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.41 - 6.08i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.81 - 8.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.593 - 1.82i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.77 + 7.95i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.62 - 4.81i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.12iT - 43T^{2} \)
47 \( 1 + (4.17 + 1.35i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.585 + 0.190i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.72 + 1.98i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.75 + 4.18i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.37 + 0.447i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.05 - 3.23i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.838 - 1.15i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.38 - 13.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.70 - 0.552i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-9.97 + 7.24i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-4.35 - 1.41i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.44009347193105619669940718868, −11.02707366043426568146520877997, −9.465664698630404275024256965795, −8.850869376033382125042040433434, −8.088590506336861871226485457187, −7.04191553163564514704684904657, −5.75048983547324883170326211288, −4.81708838114542188888182919048, −3.31742173597990731053162789600, −2.44580242967457421102962236818, 0.47576088814830292337281841183, 2.72077132395017895541507224411, 3.92634389165901819725351205734, 4.79524023821691961349572825473, 6.34196880474849359261109756121, 7.53955278325274613334677398449, 7.987118467872972238647956405481, 8.865854403732556950912942175330, 10.30895815778117831871700504035, 10.96871390942405338892990018469

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