Properties

Label 2-20e2-20.7-c5-0-7
Degree $2$
Conductor $400$
Sign $-0.880 + 0.473i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.880 + 0.473i$
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: \((2,\ 400,\ (\ :5/2),\ -0.880 + 0.473i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.691725743\)
\(L(\frac12)\) \(\approx\) \(1.691725743\)
\(L(\frac{7}{2})\) 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

−10.67362642392215354382186258509, −9.602460053389161732613628953522, −9.451036629455313888193992138312, −8.547630599330800105271304470654, −7.46240824154378365471261852167, −6.12271357407724059384550664955, −5.09887808363962835988972637390, −3.80633855567751915823463988239, −3.08571137322955083124840162511, −2.05004837482403233266245740876, 0.34848590167076951132319722845, 1.31604730590091623918576145527, 3.00224284603275603648535001666, 3.28817686927100302989038252398, 5.02699056791739881001917887584, 6.72783681320703935474451244564, 7.00522237241801538644267241086, 7.974225076720282224332775003354, 8.879706881738638102618324793523, 9.874356634069636523414323114768

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