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Properties

Label 2-20e2-80.43-c1-0-33
Degree $2$
Conductor $400$
Sign $-0.753 - 0.656i$
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  + (0.307 − 1.38i)2-s − 2.85i·3-s + (−1.81 − 0.849i)4-s + (−3.94 − 0.879i)6-s + (0.458 − 0.458i)7-s + (−1.73 + 2.23i)8-s − 5.15·9-s + (−0.492 + 0.492i)11-s + (−2.42 + 5.17i)12-s − 4.52·13-s + (−0.492 − 0.774i)14-s + (2.55 + 3.07i)16-s + (3.12 − 3.12i)17-s + (−1.58 + 7.11i)18-s + (4.04 − 4.04i)19-s + ⋯
L(s)  = 1  + (0.217 − 0.976i)2-s − 1.64i·3-s + (−0.905 − 0.424i)4-s + (−1.60 − 0.358i)6-s + (0.173 − 0.173i)7-s + (−0.611 + 0.791i)8-s − 1.71·9-s + (−0.148 + 0.148i)11-s + (−0.700 + 1.49i)12-s − 1.25·13-s + (−0.131 − 0.207i)14-s + (0.638 + 0.769i)16-s + (0.758 − 0.758i)17-s + (−0.374 + 1.67i)18-s + (0.928 − 0.928i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.753 - 0.656i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.753 - 0.656i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391526 + 1.04526i\)
\(L(\frac12)\) \(\approx\) \(0.391526 + 1.04526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.307 + 1.38i)T \)
5 \( 1 \)
good3 \( 1 + 2.85iT - 3T^{2} \)
7 \( 1 + (-0.458 + 0.458i)T - 7iT^{2} \)
11 \( 1 + (0.492 - 0.492i)T - 11iT^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
17 \( 1 + (-3.12 + 3.12i)T - 17iT^{2} \)
19 \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \)
23 \( 1 + (-1.80 - 1.80i)T + 23iT^{2} \)
29 \( 1 + (3.83 + 3.83i)T + 29iT^{2} \)
31 \( 1 + 0.139iT - 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 7.49T + 43T^{2} \)
47 \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \)
53 \( 1 + 2.75iT - 53T^{2} \)
59 \( 1 + (3.62 + 3.62i)T + 59iT^{2} \)
61 \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \)
67 \( 1 + 3.32T + 67T^{2} \)
71 \( 1 - 1.37T + 71T^{2} \)
73 \( 1 + (-2.55 + 2.55i)T - 73iT^{2} \)
79 \( 1 + 3.86T + 79T^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (-4.95 + 4.95i)T - 97iT^{2} \)
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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

−11.08427173654377770327775885277, −9.832840806225848478694983837068, −9.008045239880111646636929034360, −7.66393798515085999133233184342, −7.24449592215304301625532196429, −5.78425971921459775534072139049, −4.84514896038918786083008130251, −3.07219917756937339987201163632, −2.08737956721368498379258887093, −0.69566009956929607529011409082, 3.18471313247461926987718762468, 4.13447517329387356535499221436, 5.18543164003541475276536575764, 5.68385265531010231587453092700, 7.22152672096311847438546057556, 8.222255414865048859094630148283, 9.146766399900070783759133535373, 9.886321504366818348258476557947, 10.56330574807322931562938350159, 11.87467324682018239539246936974

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