Properties

Label 2-20e2-1.1-c5-0-8
Degree $2$
Conductor $400$
Sign $1$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.69·3-s + 10.2·7-s − 220.·9-s − 596.·11-s + 420.·13-s − 974.·17-s + 380.·19-s + 48.1·21-s + 3.54e3·23-s − 2.17e3·27-s + 5.44e3·29-s + 3.62e3·31-s − 2.79e3·33-s − 1.75e3·37-s + 1.97e3·39-s + 263.·41-s + 1.44e4·43-s + 2.34e4·47-s − 1.67e4·49-s − 4.57e3·51-s − 3.34e4·53-s + 1.78e3·57-s + 2.90e3·59-s + 2.94e4·61-s − 2.26e3·63-s + 7.16e3·67-s + 1.66e4·69-s + ⋯
L(s)  = 1  + 0.301·3-s + 0.0791·7-s − 0.909·9-s − 1.48·11-s + 0.690·13-s − 0.817·17-s + 0.241·19-s + 0.0238·21-s + 1.39·23-s − 0.574·27-s + 1.20·29-s + 0.677·31-s − 0.447·33-s − 0.210·37-s + 0.207·39-s + 0.0245·41-s + 1.18·43-s + 1.54·47-s − 0.993·49-s − 0.246·51-s − 1.63·53-s + 0.0728·57-s + 0.108·59-s + 1.01·61-s − 0.0719·63-s + 0.194·67-s + 0.420·69-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: $1$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.847317154\)
\(L(\frac12)\) \(\approx\) \(1.847317154\)
\(L(\frac{7}{2})\) 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 \( 1 - 4.69T + 243T^{2} \)
7 \( 1 - 10.2T + 1.68e4T^{2} \)
11 \( 1 + 596.T + 1.61e5T^{2} \)
13 \( 1 - 420.T + 3.71e5T^{2} \)
17 \( 1 + 974.T + 1.41e6T^{2} \)
19 \( 1 - 380.T + 2.47e6T^{2} \)
23 \( 1 - 3.54e3T + 6.43e6T^{2} \)
29 \( 1 - 5.44e3T + 2.05e7T^{2} \)
31 \( 1 - 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.75e3T + 6.93e7T^{2} \)
41 \( 1 - 263.T + 1.15e8T^{2} \)
43 \( 1 - 1.44e4T + 1.47e8T^{2} \)
47 \( 1 - 2.34e4T + 2.29e8T^{2} \)
53 \( 1 + 3.34e4T + 4.18e8T^{2} \)
59 \( 1 - 2.90e3T + 7.14e8T^{2} \)
61 \( 1 - 2.94e4T + 8.44e8T^{2} \)
67 \( 1 - 7.16e3T + 1.35e9T^{2} \)
71 \( 1 - 8.13e4T + 1.80e9T^{2} \)
73 \( 1 + 5.51e4T + 2.07e9T^{2} \)
79 \( 1 + 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5T + 3.93e9T^{2} \)
89 \( 1 - 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 6.29e4T + 8.58e9T^{2} \)
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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

−10.69554369353183405222503916178, −9.425186295405745291368215654648, −8.537024566019851074197069375099, −7.889744253355958686489075967483, −6.68910247305146790618214224245, −5.60089071097965689606642985366, −4.67875727399690029360680269840, −3.18855396952851884476004033022, −2.39066292192021466365772690294, −0.69004512996106942220035842352, 0.69004512996106942220035842352, 2.39066292192021466365772690294, 3.18855396952851884476004033022, 4.67875727399690029360680269840, 5.60089071097965689606642985366, 6.68910247305146790618214224245, 7.889744253355958686489075967483, 8.537024566019851074197069375099, 9.425186295405745291368215654648, 10.69554369353183405222503916178

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