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Properties

Label 2-20e2-5.4-c9-0-11
Degree $2$
Conductor $400$
Sign $-0.447 - 0.894i$
Analytic cond. $206.014$
Root an. cond. $14.3531$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 272. i·3-s − 1.00e4i·7-s − 5.44e4·9-s − 4.70e4·11-s + 9.36e3i·13-s − 1.08e5i·17-s − 6.65e5·19-s + 2.72e6·21-s − 5.76e5i·23-s − 9.45e6i·27-s + 2.61e6·29-s − 3.87e6·31-s − 1.28e7i·33-s − 1.41e7i·37-s − 2.54e6·39-s + ⋯
L(s)  = 1  + 1.94i·3-s − 1.57i·7-s − 2.76·9-s − 0.969·11-s + 0.0909i·13-s − 0.315i·17-s − 1.17·19-s + 3.05·21-s − 0.429i·23-s − 3.42i·27-s + 0.685·29-s − 0.754·31-s − 1.88i·33-s − 1.24i·37-s − 0.176·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(206.014\)
Root analytic conductor: \(14.3531\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :9/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.8917932603\)
\(L(\frac12)\) \(\approx\) \(0.8917932603\)
\(L(\frac{11}{2})\) 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 \( 1 \)
good3 \( 1 - 272. iT - 1.96e4T^{2} \)
7 \( 1 + 1.00e4iT - 4.03e7T^{2} \)
11 \( 1 + 4.70e4T + 2.35e9T^{2} \)
13 \( 1 - 9.36e3iT - 1.06e10T^{2} \)
17 \( 1 + 1.08e5iT - 1.18e11T^{2} \)
19 \( 1 + 6.65e5T + 3.22e11T^{2} \)
23 \( 1 + 5.76e5iT - 1.80e12T^{2} \)
29 \( 1 - 2.61e6T + 1.45e13T^{2} \)
31 \( 1 + 3.87e6T + 2.64e13T^{2} \)
37 \( 1 + 1.41e7iT - 1.29e14T^{2} \)
41 \( 1 - 4.62e6T + 3.27e14T^{2} \)
43 \( 1 + 8.31e6iT - 5.02e14T^{2} \)
47 \( 1 - 2.51e7iT - 1.11e15T^{2} \)
53 \( 1 - 3.49e7iT - 3.29e15T^{2} \)
59 \( 1 - 6.71e6T + 8.66e15T^{2} \)
61 \( 1 + 4.75e6T + 1.16e16T^{2} \)
67 \( 1 + 1.38e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.54e8T + 4.58e16T^{2} \)
73 \( 1 - 2.41e8iT - 5.88e16T^{2} \)
79 \( 1 - 2.61e8T + 1.19e17T^{2} \)
83 \( 1 - 6.55e8iT - 1.86e17T^{2} \)
89 \( 1 - 1.00e9T + 3.50e17T^{2} \)
97 \( 1 + 1.24e9iT - 7.60e17T^{2} \)
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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

−10.31467586653157265838173810009, −9.355263013679437924128772423131, −8.443690840933239442382164295261, −7.42494656927140538804045234095, −6.08509564790683321232505645827, −4.95890712187277402752800706302, −4.30425714794621860756367645641, −3.58774079487115715027195336341, −2.49091637931035581941185289538, −0.53691244650309628237494344902, 0.26635940276147155299680305571, 1.61262926726973722118267917690, 2.30766631322051468402236444973, 3.04284289830453131673823211521, 5.13309297660004196959827738388, 5.93302501674199644650458251783, 6.59375804375869854630514932722, 7.71843123294094880592981645790, 8.371360964647336348918789052566, 9.017785542217583075828169720159

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