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Properties

Label 2-20e2-25.6-c1-0-2
Degree $2$
Conductor $400$
Sign $-0.529 - 0.848i$
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  + (−2.19 + 1.59i)3-s + (0.548 + 2.16i)5-s + 4.40·7-s + (1.34 − 4.15i)9-s + (0.516 + 1.58i)11-s + (−0.795 + 2.44i)13-s + (−4.66 − 3.88i)15-s + (−0.239 − 0.173i)17-s + (−3.13 − 2.28i)19-s + (−9.66 + 7.02i)21-s + (2.71 + 8.36i)23-s + (−4.39 + 2.37i)25-s + (1.14 + 3.52i)27-s + (−0.177 + 0.129i)29-s + (−5.90 − 4.29i)31-s + ⋯
L(s)  = 1  + (−1.26 + 0.921i)3-s + (0.245 + 0.969i)5-s + 1.66·7-s + (0.449 − 1.38i)9-s + (0.155 + 0.478i)11-s + (−0.220 + 0.679i)13-s + (−1.20 − 1.00i)15-s + (−0.0579 − 0.0421i)17-s + (−0.720 − 0.523i)19-s + (−2.10 + 1.53i)21-s + (0.566 + 1.74i)23-s + (−0.879 + 0.475i)25-s + (0.220 + 0.678i)27-s + (−0.0329 + 0.0239i)29-s + (−1.06 − 0.771i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.529 - 0.848i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.529 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489563 + 0.882710i\)
\(L(\frac12)\) \(\approx\) \(0.489563 + 0.882710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.548 - 2.16i)T \)
good3 \( 1 + (2.19 - 1.59i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 - 4.40T + 7T^{2} \)
11 \( 1 + (-0.516 - 1.58i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.795 - 2.44i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.239 + 0.173i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.13 + 2.28i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.71 - 8.36i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.177 - 0.129i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.90 + 4.29i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.231 - 0.711i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.947 - 2.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.913T + 43T^{2} \)
47 \( 1 + (-0.00570 + 0.00414i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.20 + 6.68i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.635 - 1.95i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.52 + 10.8i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-4.94 - 3.59i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (6.51 - 4.73i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.08 - 3.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.42 + 6.84i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.30 - 1.67i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.02 + 3.16i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.73 - 6.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.31506098651296269307670260105, −10.96076668071952366826608809742, −10.00269062860857646984487502063, −9.138665165480040053626193381375, −7.67611383476608770858154101606, −6.77988226441881100123373212033, −5.61712103280120050571422584123, −4.86628235708050616583709675548, −3.91319614222806791580224939632, −1.93501076475691125092376705420, 0.820305264946501133612021259195, 1.94327408557532789611753685901, 4.46070195035396980810975051770, 5.27124311631413628922263690876, 5.97188372064384007313466676469, 7.19186062739387840177900305396, 8.181131893825859751181472983670, 8.816894514252280653518310502621, 10.55524836811569140112690460257, 10.98615250021305925745469933407

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