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Properties

Label 2-20e2-25.11-c1-0-5
Degree $2$
Conductor $400$
Sign $0.424 - 0.905i$
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  + (−0.529 + 1.63i)3-s + (2.02 + 0.951i)5-s + 2.77·7-s + (0.0483 + 0.0351i)9-s + (2.24 − 1.63i)11-s + (−4.59 − 3.33i)13-s + (−2.62 + 2.79i)15-s + (1.59 + 4.90i)17-s + (−0.436 − 1.34i)19-s + (−1.47 + 4.52i)21-s + (−0.529 + 0.384i)23-s + (3.19 + 3.84i)25-s + (−4.24 + 3.08i)27-s + (1.26 − 3.89i)29-s + (2.20 + 6.77i)31-s + ⋯
L(s)  = 1  + (−0.305 + 0.941i)3-s + (0.905 + 0.425i)5-s + 1.04·7-s + (0.0161 + 0.0117i)9-s + (0.676 − 0.491i)11-s + (−1.27 − 0.925i)13-s + (−0.677 + 0.722i)15-s + (0.386 + 1.18i)17-s + (−0.100 − 0.308i)19-s + (−0.320 + 0.987i)21-s + (−0.110 + 0.0802i)23-s + (0.638 + 0.769i)25-s + (−0.816 + 0.593i)27-s + (0.235 − 0.723i)29-s + (0.395 + 1.21i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.424 - 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34709 + 0.855843i\)
\(L(\frac12)\) \(\approx\) \(1.34709 + 0.855843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 - 0.951i)T \)
good3 \( 1 + (0.529 - 1.63i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 - 2.77T + 7T^{2} \)
11 \( 1 + (-2.24 + 1.63i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (4.59 + 3.33i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.59 - 4.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.436 + 1.34i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.529 - 0.384i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.26 + 3.89i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.20 - 6.77i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.847 - 0.615i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.36 + 5.35i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.24T + 43T^{2} \)
47 \( 1 + (-0.857 + 2.63i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.162 - 0.500i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.05 + 2.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.76 + 6.36i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.33 - 4.11i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.09 + 12.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.40 - 2.47i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-3.05 + 9.41i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.44 + 4.44i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-7.43 + 5.39i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (0.0278 - 0.0857i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.20286316352636011569320311135, −10.30945144433596308803767603941, −10.02418355320716842158940466873, −8.812855788953534337100958806257, −7.81386279605098136043262431012, −6.57160538650483856549821850210, −5.41273370041540949537307165422, −4.79691297656726588680002525429, −3.41706254268554446802151517874, −1.82918853628971664033885878348, 1.32725420796002826114014424049, 2.25821283729229613438275794444, 4.47111093080161753879073364864, 5.26931115940277519240642237016, 6.54873400138362710253518130374, 7.19734900958085937213747477358, 8.233609058754643439971062230901, 9.452079308390581065208808491194, 9.953845198250042448780757371395, 11.53715641204274832701190585321

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