Properties

Label 2-20e2-5.4-c9-0-14
Degree $2$
Conductor $400$
Sign $0.894 + 0.447i$
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  − 268. i·3-s − 637. i·7-s − 5.24e4·9-s + 4.90e4·11-s − 7.27e4i·13-s + 6.73e4i·17-s + 3.41e5·19-s − 1.71e5·21-s − 1.34e5i·23-s + 8.81e6i·27-s − 4.45e6·29-s − 4.56e5·31-s − 1.31e7i·33-s + 1.30e7i·37-s − 1.95e7·39-s + ⋯
L(s)  = 1  − 1.91i·3-s − 0.100i·7-s − 2.66·9-s + 1.00·11-s − 0.706i·13-s + 0.195i·17-s + 0.600·19-s − 0.192·21-s − 0.100i·23-s + 3.19i·27-s − 1.17·29-s − 0.0887·31-s − 1.93i·33-s + 1.14i·37-s − 1.35·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
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: \((2,\ 400,\ (\ :9/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.507445123\)
\(L(\frac12)\) \(\approx\) \(1.507445123\)
\(L(\frac{11}{2})\) 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 \( 1 + 268. iT - 1.96e4T^{2} \)
7 \( 1 + 637. iT - 4.03e7T^{2} \)
11 \( 1 - 4.90e4T + 2.35e9T^{2} \)
13 \( 1 + 7.27e4iT - 1.06e10T^{2} \)
17 \( 1 - 6.73e4iT - 1.18e11T^{2} \)
19 \( 1 - 3.41e5T + 3.22e11T^{2} \)
23 \( 1 + 1.34e5iT - 1.80e12T^{2} \)
29 \( 1 + 4.45e6T + 1.45e13T^{2} \)
31 \( 1 + 4.56e5T + 2.64e13T^{2} \)
37 \( 1 - 1.30e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.56e7T + 3.27e14T^{2} \)
43 \( 1 - 3.42e6iT - 5.02e14T^{2} \)
47 \( 1 - 3.39e7iT - 1.11e15T^{2} \)
53 \( 1 - 8.42e7iT - 3.29e15T^{2} \)
59 \( 1 + 7.46e7T + 8.66e15T^{2} \)
61 \( 1 - 1.78e8T + 1.16e16T^{2} \)
67 \( 1 + 6.94e7iT - 2.72e16T^{2} \)
71 \( 1 - 2.07e8T + 4.58e16T^{2} \)
73 \( 1 - 3.02e8iT - 5.88e16T^{2} \)
79 \( 1 - 3.72e8T + 1.19e17T^{2} \)
83 \( 1 - 4.50e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.82e7T + 3.50e17T^{2} \)
97 \( 1 - 7.85e8iT - 7.60e17T^{2} \)
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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

−9.479106613956452758092287958374, −8.490670968702649639534258527202, −7.73837263274847568635099043319, −6.94314978380085602795139991253, −6.20537245415881483520461039812, −5.30367736508203334278249856881, −3.59781302693165884322551604036, −2.54902659804746315498872973674, −1.46571376687663524340359609382, −0.857807587561656437438427072657, 0.32263862312116705775903853131, 2.03464283437350016911880813373, 3.44441584508934158966610573332, 3.94310635609647069876504552275, 4.97020286264555513495563663171, 5.74066860903933559754492245271, 6.95088505384704750586739579898, 8.418111301598675490343687206134, 9.227433952898093371891930564424, 9.659480280245259221561462774740

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