Properties

Degree $2$
Conductor $400$
Sign $1$
Motivic weight $5$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.52·3-s + 68.9·7-s − 212.·9-s + 486.·11-s + 428.·13-s − 1.80e3·17-s + 1.04e3·19-s − 380.·21-s + 686.·23-s + 2.51e3·27-s − 1.33e3·29-s − 7.99e3·31-s − 2.68e3·33-s + 1.97e3·37-s − 2.36e3·39-s + 1.07e4·41-s + 1.50e4·43-s + 895.·47-s − 1.20e4·49-s + 9.94e3·51-s + 1.93e4·53-s − 5.78e3·57-s − 2.11e4·59-s − 2.77e4·61-s − 1.46e4·63-s − 7.71e3·67-s − 3.79e3·69-s + ⋯
L(s)  = 1  − 0.354·3-s + 0.531·7-s − 0.874·9-s + 1.21·11-s + 0.703·13-s − 1.51·17-s + 0.665·19-s − 0.188·21-s + 0.270·23-s + 0.664·27-s − 0.295·29-s − 1.49·31-s − 0.429·33-s + 0.236·37-s − 0.249·39-s + 1.00·41-s + 1.23·43-s + 0.0591·47-s − 0.717·49-s + 0.535·51-s + 0.945·53-s − 0.235·57-s − 0.792·59-s − 0.953·61-s − 0.465·63-s − 0.210·67-s − 0.0959·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(5\)
Character: $\chi_{400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.866688079\)
\(L(\frac12)\) \(\approx\) \(1.866688079\)
\(L(\frac{7}{2})\) 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 \( 1 + 5.52T + 243T^{2} \)
7 \( 1 - 68.9T + 1.68e4T^{2} \)
11 \( 1 - 486.T + 1.61e5T^{2} \)
13 \( 1 - 428.T + 3.71e5T^{2} \)
17 \( 1 + 1.80e3T + 1.41e6T^{2} \)
19 \( 1 - 1.04e3T + 2.47e6T^{2} \)
23 \( 1 - 686.T + 6.43e6T^{2} \)
29 \( 1 + 1.33e3T + 2.05e7T^{2} \)
31 \( 1 + 7.99e3T + 2.86e7T^{2} \)
37 \( 1 - 1.97e3T + 6.93e7T^{2} \)
41 \( 1 - 1.07e4T + 1.15e8T^{2} \)
43 \( 1 - 1.50e4T + 1.47e8T^{2} \)
47 \( 1 - 895.T + 2.29e8T^{2} \)
53 \( 1 - 1.93e4T + 4.18e8T^{2} \)
59 \( 1 + 2.11e4T + 7.14e8T^{2} \)
61 \( 1 + 2.77e4T + 8.44e8T^{2} \)
67 \( 1 + 7.71e3T + 1.35e9T^{2} \)
71 \( 1 - 5.14e4T + 1.80e9T^{2} \)
73 \( 1 - 4.37e4T + 2.07e9T^{2} \)
79 \( 1 - 6.22e3T + 3.07e9T^{2} \)
83 \( 1 - 5.29e4T + 3.93e9T^{2} \)
89 \( 1 - 4.46e4T + 5.58e9T^{2} \)
97 \( 1 - 1.48e5T + 8.58e9T^{2} \)
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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

−10.89076753273590244511245547042, −9.238665763092530542728124660557, −8.867865931595605439979241255843, −7.65835145371160270939437195432, −6.54663478839393067463603638661, −5.75541962469641083996100860404, −4.60557277768985972496482298225, −3.52202524990560935427513190270, −2.04214349764026624114689933067, −0.74531022172690809737111640214, 0.74531022172690809737111640214, 2.04214349764026624114689933067, 3.52202524990560935427513190270, 4.60557277768985972496482298225, 5.75541962469641083996100860404, 6.54663478839393067463603638661, 7.65835145371160270939437195432, 8.867865931595605439979241255843, 9.238665763092530542728124660557, 10.89076753273590244511245547042

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