Properties

Label 2-20e2-5.4-c5-0-15
Degree $2$
Conductor $400$
Sign $-0.447 - 0.894i$
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  + 4i·3-s + 192i·7-s + 227·9-s + 148·11-s + 286i·13-s + 1.67e3i·17-s + 1.06e3·19-s − 768·21-s − 2.97e3i·23-s + 1.88e3i·27-s + 3.41e3·29-s + 2.44e3·31-s + 592i·33-s − 182i·37-s − 1.14e3·39-s + ⋯
L(s)  = 1  + 0.256i·3-s + 1.48i·7-s + 0.934·9-s + 0.368·11-s + 0.469i·13-s + 1.40i·17-s + 0.673·19-s − 0.380·21-s − 1.17i·23-s + 0.496i·27-s + 0.752·29-s + 0.457·31-s + 0.0946i·33-s − 0.0218i·37-s − 0.120·39-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: $-0.447 - 0.894i$
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: \((2,\ 400,\ (\ :5/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.171368568\)
\(L(\frac12)\) \(\approx\) \(2.171368568\)
\(L(\frac{7}{2})\) 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 \( 1 - 4iT - 243T^{2} \)
7 \( 1 - 192iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 - 286iT - 3.71e5T^{2} \)
17 \( 1 - 1.67e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.06e3T + 2.47e6T^{2} \)
23 \( 1 + 2.97e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.41e3T + 2.05e7T^{2} \)
31 \( 1 - 2.44e3T + 2.86e7T^{2} \)
37 \( 1 + 182iT - 6.93e7T^{2} \)
41 \( 1 + 9.39e3T + 1.15e8T^{2} \)
43 \( 1 - 1.24e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.20e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.38e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.00e4T + 7.14e8T^{2} \)
61 \( 1 - 3.23e4T + 8.44e8T^{2} \)
67 \( 1 - 6.09e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.26e4T + 1.80e9T^{2} \)
73 \( 1 + 3.87e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.33e4T + 3.07e9T^{2} \)
83 \( 1 + 1.67e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.01e5T + 5.58e9T^{2} \)
97 \( 1 - 1.19e5iT - 8.58e9T^{2} \)
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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

−10.63711314705327424189992972677, −9.826582448689659733834801006543, −8.904592511014022302758551925819, −8.229087475235888577866423995318, −6.85172931810763192178561952344, −6.02540750850544426033913283284, −4.91703326635284201664069702733, −3.88356738675044604493213960324, −2.51317803496221983519859931273, −1.37770476687946560940805465700, 0.58035348156098161694013137955, 1.42773789144470423649753859643, 3.15400725806146634871805935912, 4.20066058815643123634873043310, 5.18436907243529206725369071465, 6.71575578682867340735517279626, 7.26370010723787706581985549315, 8.056006419736228498335078704124, 9.564058936778676808309358684555, 10.02358959797624898560643994627

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