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Properties

Label 2-20e2-100.3-c1-0-5
Degree $2$
Conductor $400$
Sign $-0.0973 + 0.995i$
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  + (−2.49 + 0.395i)3-s + (−0.616 + 2.14i)5-s + (−1.08 − 1.08i)7-s + (3.21 − 1.04i)9-s + (−0.665 − 0.216i)11-s + (−0.606 + 1.18i)13-s + (0.687 − 5.60i)15-s + (0.880 − 5.55i)17-s + (1.31 − 0.952i)19-s + (3.12 + 2.26i)21-s + (−3.79 − 7.44i)23-s + (−4.24 − 2.64i)25-s + (−0.866 + 0.441i)27-s + (4.72 − 6.49i)29-s + (2.81 + 3.86i)31-s + ⋯
L(s)  = 1  + (−1.44 + 0.228i)3-s + (−0.275 + 0.961i)5-s + (−0.408 − 0.408i)7-s + (1.07 − 0.348i)9-s + (−0.200 − 0.0651i)11-s + (−0.168 + 0.329i)13-s + (0.177 − 1.44i)15-s + (0.213 − 1.34i)17-s + (0.300 − 0.218i)19-s + (0.681 + 0.495i)21-s + (−0.791 − 1.55i)23-s + (−0.848 − 0.529i)25-s + (−0.166 + 0.0849i)27-s + (0.876 − 1.20i)29-s + (0.504 + 0.694i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0973 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.0973 + 0.995i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.0973 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233666 - 0.257640i\)
\(L(\frac12)\) \(\approx\) \(0.233666 - 0.257640i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.616 - 2.14i)T \)
good3 \( 1 + (2.49 - 0.395i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (1.08 + 1.08i)T + 7iT^{2} \)
11 \( 1 + (0.665 + 0.216i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.606 - 1.18i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.880 + 5.55i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-1.31 + 0.952i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.79 + 7.44i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-4.72 + 6.49i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.81 - 3.86i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (9.91 + 5.05i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (0.0489 + 0.150i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (5.11 - 5.11i)T - 43iT^{2} \)
47 \( 1 + (-1.04 - 6.62i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.570 - 3.60i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (4.70 + 14.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.50 - 7.72i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.12 - 0.969i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-0.603 + 0.830i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.39 + 2.23i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-8.82 - 6.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.609 - 3.84i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (8.25 + 2.68i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.75 + 0.595i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.99055000571185875173212682447, −10.35229854872712314708373308392, −9.643450257394867100603978926395, −8.061520406741883848283584663174, −6.84751611760471592053043702189, −6.46897116180834562002085620050, −5.23141203022142712837014440215, −4.23147582909714846690866879822, −2.78705178428651710968870906566, −0.28428724799524653400888120113, 1.44153800137238959290139152356, 3.66206197244882624926410440645, 5.06259763092780618301120947906, 5.62635332465061884624043940120, 6.56529602068143274254374525636, 7.78410550052109381459313913288, 8.723601410559526022483717425559, 9.918100760766983615281500318415, 10.65537294919874341954473615957, 11.88169938686298064537821247899

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