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Properties

Label 2-20e2-5.4-c5-0-0
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  + 28.7i·3-s − 42.1i·7-s − 581.·9-s − 416.·11-s + 966. i·13-s + 1.83e3i·17-s + 317.·19-s + 1.21e3·21-s − 1.56e3i·23-s − 9.72e3i·27-s − 7.75e3·29-s − 102.·31-s − 1.19e4i·33-s − 1.93e3i·37-s − 2.77e4·39-s + ⋯
L(s)  = 1  + 1.84i·3-s − 0.325i·7-s − 2.39·9-s − 1.03·11-s + 1.58i·13-s + 1.53i·17-s + 0.201·19-s + 0.598·21-s − 0.618i·23-s − 2.56i·27-s − 1.71·29-s − 0.0191·31-s − 1.91i·33-s − 0.232i·37-s − 2.92·39-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

\(L(3)\) \(\approx\) \(0.1923740517\)
\(L(\frac12)\) \(\approx\) \(0.1923740517\)
\(L(\frac{7}{2})\) 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 \( 1 - 28.7iT - 243T^{2} \)
7 \( 1 + 42.1iT - 1.68e4T^{2} \)
11 \( 1 + 416.T + 1.61e5T^{2} \)
13 \( 1 - 966. iT - 3.71e5T^{2} \)
17 \( 1 - 1.83e3iT - 1.41e6T^{2} \)
19 \( 1 - 317.T + 2.47e6T^{2} \)
23 \( 1 + 1.56e3iT - 6.43e6T^{2} \)
29 \( 1 + 7.75e3T + 2.05e7T^{2} \)
31 \( 1 + 102.T + 2.86e7T^{2} \)
37 \( 1 + 1.93e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.99e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.86e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.49e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.98e4T + 7.14e8T^{2} \)
61 \( 1 + 1.80e4T + 8.44e8T^{2} \)
67 \( 1 + 5.50e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.12e4T + 1.80e9T^{2} \)
73 \( 1 + 4.01e3iT - 2.07e9T^{2} \)
79 \( 1 - 2.40e4T + 3.07e9T^{2} \)
83 \( 1 + 7.05e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.07e4T + 5.58e9T^{2} \)
97 \( 1 - 3.11e4iT - 8.58e9T^{2} \)
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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

−10.80114912694782957916878909592, −10.46106763927210967966829808652, −9.413630066762025771708467308647, −8.821213200438441095711948009852, −7.70122776169152622767157634266, −6.21714727104239879680102179070, −5.23454829926243579760615188934, −4.25510654705136452631931707885, −3.63285294921867625123270432616, −2.17472251969225840768940775764, 0.05286702441760970953360181297, 0.980261759313446434123884768672, 2.36164130550640247167346062190, 3.06889897384129707026770485874, 5.33147719771114408454969296635, 5.79422326553415760912951776346, 7.19793958871723590049513432859, 7.62023001759786204767753525955, 8.429924348766463160351429576065, 9.595356569792380910029991152477

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