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Properties

Label 2-20e2-20.7-c5-0-43
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  + (−0.839 − 0.839i)3-s + (99.3 − 99.3i)7-s − 241. i·9-s − 637. i·11-s + (640. − 640. i)13-s + (−648. − 648. i)17-s − 2.50e3·19-s − 166.·21-s + (2.80e3 + 2.80e3i)23-s + (−406. + 406. i)27-s + 4.95e3i·29-s − 1.96e3i·31-s + (−535. + 535. i)33-s + (1.89e3 + 1.89e3i)37-s − 1.07e3·39-s + ⋯
L(s)  = 1  + (−0.0538 − 0.0538i)3-s + (0.766 − 0.766i)7-s − 0.994i·9-s − 1.58i·11-s + (1.05 − 1.05i)13-s + (−0.544 − 0.544i)17-s − 1.59·19-s − 0.0825·21-s + (1.10 + 1.10i)23-s + (−0.107 + 0.107i)27-s + 1.09i·29-s − 0.366i·31-s + (−0.0856 + 0.0856i)33-s + (0.227 + 0.227i)37-s − 0.113·39-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

\(L(3)\) \(\approx\) \(1.726628183\)
\(L(\frac12)\) \(\approx\) \(1.726628183\)
\(L(\frac{7}{2})\) 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 \( 1 + (0.839 + 0.839i)T + 243iT^{2} \)
7 \( 1 + (-99.3 + 99.3i)T - 1.68e4iT^{2} \)
11 \( 1 + 637. iT - 1.61e5T^{2} \)
13 \( 1 + (-640. + 640. i)T - 3.71e5iT^{2} \)
17 \( 1 + (648. + 648. i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.50e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.80e3 - 2.80e3i)T + 6.43e6iT^{2} \)
29 \( 1 - 4.95e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.96e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.89e3 - 1.89e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 5.82e3T + 1.15e8T^{2} \)
43 \( 1 + (1.06e4 + 1.06e4i)T + 1.47e8iT^{2} \)
47 \( 1 + (-8.30e3 + 8.30e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (7.43e3 - 7.43e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 1.67e4T + 7.14e8T^{2} \)
61 \( 1 - 2.37e4T + 8.44e8T^{2} \)
67 \( 1 + (-4.15e3 + 4.15e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 1.22e4iT - 1.80e9T^{2} \)
73 \( 1 + (3.60e4 - 3.60e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 6.43e4T + 3.07e9T^{2} \)
83 \( 1 + (6.28e4 + 6.28e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 2.44e4iT - 5.58e9T^{2} \)
97 \( 1 + (-8.96e4 - 8.96e4i)T + 8.58e9iT^{2} \)
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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

−10.31509907871545793183008008242, −8.825012700908386402704112699138, −8.478464578982571325920739840637, −7.22094981562469485072481023408, −6.25385885404830073533581833791, −5.31276164020093146498209098185, −3.93929754333693661057082010829, −3.13324420624774942100692598699, −1.27722535194451609479649153366, −0.43841097996065122671556964886, 1.74013426726122584695134011877, 2.29968783100627977655519218056, 4.32405524978016896476077235745, 4.77228494863977814441326659891, 6.15747947927717716855752906992, 7.05159380824083816987368557383, 8.327191590978422286668480274714, 8.770921289639734508989603379033, 10.02911867784283746804634438051, 10.93352530710257088221668164268

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