Properties

Label 2-20e2-5.4-c3-0-15
Degree $2$
Conductor $400$
Sign $0.447 + 0.894i$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
Motivic weight $3$
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 − 16i·7-s + 11·9-s + 60·11-s + 86i·13-s − 18i·17-s + 44·19-s − 64·21-s − 48i·23-s − 152i·27-s + 186·29-s − 176·31-s − 240i·33-s − 254i·37-s + 344·39-s + ⋯
L(s)  = 1  − 0.769i·3-s − 0.863i·7-s + 0.407·9-s + 1.64·11-s + 1.83i·13-s − 0.256i·17-s + 0.531·19-s − 0.665·21-s − 0.435i·23-s − 1.08i·27-s + 1.19·29-s − 1.01·31-s − 1.26i·33-s − 1.12i·37-s + 1.41·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.238190706\)
\(L(\frac12)\) \(\approx\) \(2.238190706\)
\(L(\frac{5}{2})\) 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 \( 1 + 4iT - 27T^{2} \)
7 \( 1 + 16iT - 343T^{2} \)
11 \( 1 - 60T + 1.33e3T^{2} \)
13 \( 1 - 86iT - 2.19e3T^{2} \)
17 \( 1 + 18iT - 4.91e3T^{2} \)
19 \( 1 - 44T + 6.85e3T^{2} \)
23 \( 1 + 48iT - 1.21e4T^{2} \)
29 \( 1 - 186T + 2.43e4T^{2} \)
31 \( 1 + 176T + 2.97e4T^{2} \)
37 \( 1 + 254iT - 5.06e4T^{2} \)
41 \( 1 - 186T + 6.89e4T^{2} \)
43 \( 1 - 100iT - 7.95e4T^{2} \)
47 \( 1 - 168iT - 1.03e5T^{2} \)
53 \( 1 + 498iT - 1.48e5T^{2} \)
59 \( 1 + 252T + 2.05e5T^{2} \)
61 \( 1 + 58T + 2.26e5T^{2} \)
67 \( 1 + 1.03e3iT - 3.00e5T^{2} \)
71 \( 1 + 168T + 3.57e5T^{2} \)
73 \( 1 - 506iT - 3.89e5T^{2} \)
79 \( 1 - 272T + 4.93e5T^{2} \)
83 \( 1 + 948iT - 5.71e5T^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 766iT - 9.12e5T^{2} \)
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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

−10.83168962650444378522874956162, −9.595669822183827216900679332762, −9.003198762170128608930132210084, −7.63043403653698445783749631998, −6.85758040629342503314450057238, −6.37189256320536165246887805706, −4.54473166964225574307258386249, −3.81115020879221736491172020123, −1.91250631198220828678566325703, −0.948532497873303986628458095219, 1.22841051218494087435189730145, 3.00463255664906067961686268508, 4.00878297577036044080831815760, 5.19424067709232827279617385093, 6.07069903855916543696095651768, 7.28953775353415661464669081387, 8.470641273346830992127916313401, 9.278612194243327085820332517192, 10.03757263700532109462940454548, 10.88553657362914629775673542227

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