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Properties

Label 2-20e2-20.7-c5-0-0
Degree $2$
Conductor $400$
Sign $0.0299 - 0.999i$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−17.2 − 17.2i)3-s + (154. − 154. i)7-s + 355. i·9-s − 127. i·11-s + (−335. + 335. i)13-s + (1.15e3 + 1.15e3i)17-s − 28.2·19-s − 5.32e3·21-s + (−2.78e3 − 2.78e3i)23-s + (1.93e3 − 1.93e3i)27-s − 3.38e3i·29-s + 5.38e3i·31-s + (−2.20e3 + 2.20e3i)33-s + (−1.15e4 − 1.15e4i)37-s + 1.15e4·39-s + ⋯
L(s)  = 1  + (−1.10 − 1.10i)3-s + (1.18 − 1.18i)7-s + 1.46i·9-s − 0.317i·11-s + (−0.549 + 0.549i)13-s + (0.969 + 0.969i)17-s − 0.0179·19-s − 2.63·21-s + (−1.09 − 1.09i)23-s + (0.511 − 0.511i)27-s − 0.748i·29-s + 1.00i·31-s + (−0.352 + 0.352i)33-s + (−1.38 − 1.38i)37-s + 1.21·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.0299 - 0.999i$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 0.0299 - 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.04089763183\)
\(L(\frac12)\) \(\approx\) \(0.04089763183\)
\(L(\frac{7}{2})\) 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 \( 1 + (17.2 + 17.2i)T + 243iT^{2} \)
7 \( 1 + (-154. + 154. i)T - 1.68e4iT^{2} \)
11 \( 1 + 127. iT - 1.61e5T^{2} \)
13 \( 1 + (335. - 335. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.15e3 - 1.15e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 28.2T + 2.47e6T^{2} \)
23 \( 1 + (2.78e3 + 2.78e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 3.38e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.38e3iT - 2.86e7T^{2} \)
37 \( 1 + (1.15e4 + 1.15e4i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.11e4T + 1.15e8T^{2} \)
43 \( 1 + (1.43e3 + 1.43e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (219. - 219. i)T - 2.29e8iT^{2} \)
53 \( 1 + (2.27e4 - 2.27e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.21e4T + 7.14e8T^{2} \)
61 \( 1 + 1.43e3T + 8.44e8T^{2} \)
67 \( 1 + (2.89e4 - 2.89e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.41e4iT - 1.80e9T^{2} \)
73 \( 1 + (2.85e4 - 2.85e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 2.34e4T + 3.07e9T^{2} \)
83 \( 1 + (1.89e4 + 1.89e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 8.17e3iT - 5.58e9T^{2} \)
97 \( 1 + (-7.67e4 - 7.67e4i)T + 8.58e9iT^{2} \)
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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

−10.72272115206417834548214215308, −10.25852902746218238933781866150, −8.493709698894788714532783319805, −7.67426136272587607851177789062, −6.98566934602492282375036319339, −6.01687298022187337238714772525, −5.02226410295616191917507513295, −3.96071923235385768052223769182, −1.91758163423716022638599496973, −1.11082329599322566100158986854, 0.01270343749286886269451444490, 1.74244284991654380808247478001, 3.29993723546061885998534498969, 4.80090064612268766641753645935, 5.19320303092703487713723175557, 5.97850157504590978565684439961, 7.49020542893792155190565988758, 8.489083616895654168373888473504, 9.660715999314904874874012155007, 10.14872242145102315254071441051

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