Properties

Degree $2$
Conductor $400$
Sign $-0.447 - 0.894i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22i·3-s + 218i·7-s − 241·9-s + 480·11-s − 622i·13-s − 186i·17-s − 1.20e3·19-s + 4.79e3·21-s + 3.18e3i·23-s − 44i·27-s − 5.52e3·29-s − 9.35e3·31-s − 1.05e4i·33-s − 5.61e3i·37-s − 1.36e4·39-s + ⋯
L(s)  = 1  − 1.41i·3-s + 1.68i·7-s − 0.991·9-s + 1.19·11-s − 1.02i·13-s − 0.156i·17-s − 0.765·19-s + 2.37·21-s + 1.25i·23-s − 0.0116i·27-s − 1.22·29-s − 1.74·31-s − 1.68i·33-s − 0.674i·37-s − 1.44·39-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(3)\) \(\approx\) \(0.2976329100\)
\(L(\frac12)\) \(\approx\) \(0.2976329100\)
\(L(\frac{7}{2})\) 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 \( 1 + 22iT - 243T^{2} \)
7 \( 1 - 218iT - 1.68e4T^{2} \)
11 \( 1 - 480T + 1.61e5T^{2} \)
13 \( 1 + 622iT - 3.71e5T^{2} \)
17 \( 1 + 186iT - 1.41e6T^{2} \)
19 \( 1 + 1.20e3T + 2.47e6T^{2} \)
23 \( 1 - 3.18e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.52e3T + 2.05e7T^{2} \)
31 \( 1 + 9.35e3T + 2.86e7T^{2} \)
37 \( 1 + 5.61e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.43e4T + 1.15e8T^{2} \)
43 \( 1 - 370iT - 1.47e8T^{2} \)
47 \( 1 - 1.61e4iT - 2.29e8T^{2} \)
53 \( 1 + 4.37e3iT - 4.18e8T^{2} \)
59 \( 1 + 1.17e4T + 7.14e8T^{2} \)
61 \( 1 - 1.32e4T + 8.44e8T^{2} \)
67 \( 1 + 1.15e4iT - 1.35e9T^{2} \)
71 \( 1 - 2.95e4T + 1.80e9T^{2} \)
73 \( 1 - 3.36e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.12e4T + 3.07e9T^{2} \)
83 \( 1 - 3.84e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.19e5T + 5.58e9T^{2} \)
97 \( 1 + 9.46e4iT - 8.58e9T^{2} \)
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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

−11.14180002977974230286346796683, −9.534422627379939472138700865798, −8.832618319384428701695857030777, −7.931649351082568095211837178263, −7.01030291017772773675187342935, −6.01921398360381519634105701257, −5.40689053955335448399949534211, −3.52942750425916199681839269115, −2.24357545680527458964554055139, −1.46392406625239292460273680500, 0.06969172079413995811069384016, 1.64306845570803581816906740911, 3.71088382271684895858455575310, 4.01079209902903751165275080001, 4.91454334050867966667843613230, 6.46359565226431338626738630584, 7.20787018801093003844439942436, 8.639848101825814351483785904487, 9.393117155538108805360097274862, 10.22667512986123092027863206468

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