Properties

Label 2-20e2-400.227-c1-0-32
Degree $2$
Conductor $400$
Sign $-0.665 + 0.746i$
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  + (−1.23 − 0.683i)2-s + (−0.203 − 0.0662i)3-s + (1.06 + 1.69i)4-s + (−1.71 + 1.43i)5-s + (0.207 + 0.221i)6-s + (0.740 + 0.740i)7-s + (−0.165 − 2.82i)8-s + (−2.38 − 1.73i)9-s + (3.10 − 0.611i)10-s + (−3.18 − 0.504i)11-s + (−0.105 − 0.415i)12-s + (0.927 + 0.673i)13-s + (−0.410 − 1.42i)14-s + (0.444 − 0.179i)15-s + (−1.72 + 3.60i)16-s + (3.08 − 6.06i)17-s + ⋯
L(s)  = 1  + (−0.875 − 0.482i)2-s + (−0.117 − 0.0382i)3-s + (0.533 + 0.845i)4-s + (−0.765 + 0.643i)5-s + (0.0846 + 0.0903i)6-s + (0.279 + 0.279i)7-s + (−0.0586 − 0.998i)8-s + (−0.796 − 0.578i)9-s + (0.981 − 0.193i)10-s + (−0.960 − 0.152i)11-s + (−0.0304 − 0.119i)12-s + (0.257 + 0.186i)13-s + (−0.109 − 0.380i)14-s + (0.114 − 0.0464i)15-s + (−0.430 + 0.902i)16-s + (0.748 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.665 + 0.746i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.665 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157187 - 0.351001i\)
\(L(\frac12)\) \(\approx\) \(0.157187 - 0.351001i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 + 0.683i)T \)
5 \( 1 + (1.71 - 1.43i)T \)
good3 \( 1 + (0.203 + 0.0662i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.740 - 0.740i)T + 7iT^{2} \)
11 \( 1 + (3.18 + 0.504i)T + (10.4 + 3.39i)T^{2} \)
13 \( 1 + (-0.927 - 0.673i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.08 + 6.06i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-1.53 + 3.01i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-1.28 + 8.13i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (1.07 + 2.10i)T + (-17.0 + 23.4i)T^{2} \)
31 \( 1 + (8.74 - 2.84i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.943 - 0.685i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.17 - 7.12i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.67T + 43T^{2} \)
47 \( 1 + (1.88 + 3.69i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-4.22 - 1.37i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (14.6 - 2.31i)T + (56.1 - 18.2i)T^{2} \)
61 \( 1 + (-13.0 - 2.06i)T + (58.0 + 18.8i)T^{2} \)
67 \( 1 + (1.39 + 4.30i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.332 + 1.02i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.05 - 0.483i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.62 + 11.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.90 - 2.24i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-13.0 + 9.50i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (12.4 - 6.33i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.03830406348381250312471008318, −10.15443482881966651177079686561, −9.041890638547056743078258129577, −8.283727346097529308997069797995, −7.38623448461541357912113871780, −6.52222471565364609365537428603, −5.03080506431202728841337048868, −3.37660393913411158791495194784, −2.62158752335886302459319502146, −0.33641838689042174869088363124, 1.61052676767148412946757963643, 3.55634395634767896272677940573, 5.23785656976994997174984919040, 5.70455688169973380297212112417, 7.41221032710840788373681180052, 7.934463609413990153807913559946, 8.588973500348938533729783847171, 9.720901390004576062588884218319, 10.75260335745127167205169961606, 11.24986178100323654365394214033

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