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Properties

Label 2-20e2-25.21-c1-0-9
Degree $2$
Conductor $400$
Sign $-0.461 + 0.886i$
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  + (−0.559 − 0.406i)3-s + (−0.463 + 2.18i)5-s − 3.25·7-s + (−0.779 − 2.39i)9-s + (0.967 − 2.97i)11-s + (−1.87 − 5.76i)13-s + (1.14 − 1.03i)15-s + (0.772 − 0.560i)17-s + (4.69 − 3.40i)19-s + (1.82 + 1.32i)21-s + (−0.660 + 2.03i)23-s + (−4.57 − 2.02i)25-s + (−1.18 + 3.63i)27-s + (−5.01 − 3.64i)29-s + (−2.25 + 1.63i)31-s + ⋯
L(s)  = 1  + (−0.323 − 0.234i)3-s + (−0.207 + 0.978i)5-s − 1.23·7-s + (−0.259 − 0.799i)9-s + (0.291 − 0.898i)11-s + (−0.519 − 1.59i)13-s + (0.296 − 0.267i)15-s + (0.187 − 0.136i)17-s + (1.07 − 0.782i)19-s + (0.397 + 0.289i)21-s + (−0.137 + 0.423i)23-s + (−0.914 − 0.405i)25-s + (−0.227 + 0.699i)27-s + (−0.931 − 0.677i)29-s + (−0.404 + 0.293i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.461 + 0.886i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.461 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309455 - 0.509984i\)
\(L(\frac12)\) \(\approx\) \(0.309455 - 0.509984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.463 - 2.18i)T \)
good3 \( 1 + (0.559 + 0.406i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + 3.25T + 7T^{2} \)
11 \( 1 + (-0.967 + 2.97i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1.87 + 5.76i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.772 + 0.560i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.69 + 3.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.660 - 2.03i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.01 + 3.64i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.25 - 1.63i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.459 - 1.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.49 - 7.68i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 + (2.18 + 1.58i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.984 - 0.715i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.35 + 13.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.03 + 3.19i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.44 - 2.50i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-8.07 - 5.86i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.26 - 10.0i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-3.05 - 2.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-13.0 + 9.49i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.35 + 10.3i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (3.89 + 2.83i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.13416987612213528147742363623, −9.974747554074989196390142008153, −9.440205288638357776772611350959, −8.034796096163657588495404185552, −7.06535495949955787916381115419, −6.25971019694253047471131867389, −5.47893948875892568305327972932, −3.39493309142032722150656963603, −3.08192308511892265998712084779, −0.39216927924568806853498011618, 1.92701430331381630856258792600, 3.74735449522805783996655114406, 4.73316794736800819749577030680, 5.70278220843832357704282239046, 6.89955318873850635709074653014, 7.80728120620547115256823750385, 9.194695612325708877437376307584, 9.532057305061800519431917450384, 10.57454090650960445503230912808, 11.87024773307650356430620186181

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