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Properties

Label 2-20e2-5.4-c5-0-25
Degree $2$
Conductor $400$
Sign $0.894 + 0.447i$
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  + 11i·3-s − 142i·7-s + 122·9-s − 777·11-s + 884i·13-s + 27i·17-s + 1.14e3·19-s + 1.56e3·21-s − 1.85e3i·23-s + 4.01e3i·27-s + 4.92e3·29-s − 1.80e3·31-s − 8.54e3i·33-s − 1.31e4i·37-s − 9.72e3·39-s + ⋯
L(s)  = 1  + 0.705i·3-s − 1.09i·7-s + 0.502·9-s − 1.93·11-s + 1.45i·13-s + 0.0226i·17-s + 0.727·19-s + 0.772·21-s − 0.730i·23-s + 1.05i·27-s + 1.08·29-s − 0.336·31-s − 1.36i·33-s − 1.58i·37-s − 1.02·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.654252851\)
\(L(\frac12)\) \(\approx\) \(1.654252851\)
\(L(\frac{7}{2})\) 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 \( 1 - 11iT - 243T^{2} \)
7 \( 1 + 142iT - 1.68e4T^{2} \)
11 \( 1 + 777T + 1.61e5T^{2} \)
13 \( 1 - 884iT - 3.71e5T^{2} \)
17 \( 1 - 27iT - 1.41e6T^{2} \)
19 \( 1 - 1.14e3T + 2.47e6T^{2} \)
23 \( 1 + 1.85e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.92e3T + 2.05e7T^{2} \)
31 \( 1 + 1.80e3T + 2.86e7T^{2} \)
37 \( 1 + 1.31e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.51e4T + 1.15e8T^{2} \)
43 \( 1 + 7.84e3iT - 1.47e8T^{2} \)
47 \( 1 + 6.73e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.41e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.39e4T + 7.14e8T^{2} \)
61 \( 1 - 4.74e4T + 8.44e8T^{2} \)
67 \( 1 + 1.31e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.54e3T + 1.80e9T^{2} \)
73 \( 1 + 5.98e4iT - 2.07e9T^{2} \)
79 \( 1 - 7.58e4T + 3.07e9T^{2} \)
83 \( 1 + 4.62e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.05e4T + 5.58e9T^{2} \)
97 \( 1 + 1.04e5iT - 8.58e9T^{2} \)
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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

−10.38626227254478507027231720601, −9.748524593023081797754759346216, −8.612008998714434575185033845183, −7.48897735530226113829515969757, −6.83306167443312266772909931776, −5.27294511634304500062273657882, −4.49936289110200360425232591089, −3.55964466568951636797552947134, −2.10608167794618266758825608869, −0.50736730807279747250246679695, 0.901293353191483212177317457157, 2.33223491205964119807337226066, 3.13382610006393151854910150974, 5.06847744548875266628242237652, 5.58179923653710673871905419780, 6.84904516600385868738954235827, 7.947677496463740998455460505635, 8.281945032520183825543402115921, 9.809672339620648355299120855539, 10.37670743847906470872639766521

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