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Properties

Label 2-20e2-20.3-c5-0-40
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  + (20.3 − 20.3i)3-s + (76.9 + 76.9i)7-s − 588. i·9-s + 556. i·11-s + (−141. − 141. i)13-s + (477. − 477. i)17-s + 1.60e3·19-s + 3.13e3·21-s + (346. − 346. i)23-s + (−7.03e3 − 7.03e3i)27-s − 7.48e3i·29-s − 7.92e3i·31-s + (1.13e4 + 1.13e4i)33-s + (−3.32e3 + 3.32e3i)37-s − 5.76e3·39-s + ⋯
L(s)  = 1  + (1.30 − 1.30i)3-s + (0.593 + 0.593i)7-s − 2.41i·9-s + 1.38i·11-s + (−0.231 − 0.231i)13-s + (0.400 − 0.400i)17-s + 1.02·19-s + 1.55·21-s + (0.136 − 0.136i)23-s + (−1.85 − 1.85i)27-s − 1.65i·29-s − 1.48i·31-s + (1.81 + 1.81i)33-s + (−0.399 + 0.399i)37-s − 0.606·39-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: $\chi_{400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 0.0299 + 0.999i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.822104990\)
\(L(\frac12)\) \(\approx\) \(3.822104990\)
\(L(\frac{7}{2})\) 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 \( 1 + (-20.3 + 20.3i)T - 243iT^{2} \)
7 \( 1 + (-76.9 - 76.9i)T + 1.68e4iT^{2} \)
11 \( 1 - 556. iT - 1.61e5T^{2} \)
13 \( 1 + (141. + 141. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-477. + 477. i)T - 1.41e6iT^{2} \)
19 \( 1 - 1.60e3T + 2.47e6T^{2} \)
23 \( 1 + (-346. + 346. i)T - 6.43e6iT^{2} \)
29 \( 1 + 7.48e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (3.32e3 - 3.32e3i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.87e4T + 1.15e8T^{2} \)
43 \( 1 + (-8.25e3 + 8.25e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-5.09e3 - 5.09e3i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.94e4 + 1.94e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 108.T + 7.14e8T^{2} \)
61 \( 1 - 1.42e4T + 8.44e8T^{2} \)
67 \( 1 + (-2.89e4 - 2.89e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 982. iT - 1.80e9T^{2} \)
73 \( 1 + (-2.15e4 - 2.15e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 9.38e3T + 3.07e9T^{2} \)
83 \( 1 + (9.45e3 - 9.45e3i)T - 3.93e9iT^{2} \)
89 \( 1 - 8.48e3iT - 5.58e9T^{2} \)
97 \( 1 + (1.22e5 - 1.22e5i)T - 8.58e9iT^{2} \)
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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

−9.768264461967016512940941903409, −9.304037921199588405425209043744, −8.061858839419966136474952224466, −7.67879628894127057884528718805, −6.80446618543925077081420919951, −5.53609635572520092832472117365, −4.09453997503541811642408087280, −2.64537694919650143363817635350, −2.06656118011897784127910600568, −0.848320237546306944606134934816, 1.30229910416081825439820396567, 2.92201931663959453181966089677, 3.58490982999489847791656923958, 4.62621516343063588017900149173, 5.56323482428425851557337781421, 7.31508984804705669288431738284, 8.157829013915586877524737271896, 8.907959241154613345481095110138, 9.623018290903968339402378080789, 10.79770134714517046603123828305

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