Properties

Label 2-20e2-1.1-c3-0-7
Degree $2$
Conductor $400$
Sign $1$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 6·7-s − 26·9-s + 19·11-s − 12·13-s + 75·17-s + 91·19-s − 6·21-s + 174·23-s − 53·27-s − 272·29-s + 230·31-s + 19·33-s + 182·37-s − 12·39-s + 117·41-s + 372·43-s − 52·47-s − 307·49-s + 75·51-s + 402·53-s + 91·57-s − 312·59-s + 170·61-s + 156·63-s + 763·67-s + 174·69-s + ⋯
L(s)  = 1  + 0.192·3-s − 0.323·7-s − 0.962·9-s + 0.520·11-s − 0.256·13-s + 1.07·17-s + 1.09·19-s − 0.0623·21-s + 1.57·23-s − 0.377·27-s − 1.74·29-s + 1.33·31-s + 0.100·33-s + 0.808·37-s − 0.0492·39-s + 0.445·41-s + 1.31·43-s − 0.161·47-s − 0.895·49-s + 0.205·51-s + 1.04·53-s + 0.211·57-s − 0.688·59-s + 0.356·61-s + 0.311·63-s + 1.39·67-s + 0.303·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.915441632\)
\(L(\frac12)\) \(\approx\) \(1.915441632\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
7 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 - 19 T + p^{3} T^{2} \)
13 \( 1 + 12 T + p^{3} T^{2} \)
17 \( 1 - 75 T + p^{3} T^{2} \)
19 \( 1 - 91 T + p^{3} T^{2} \)
23 \( 1 - 174 T + p^{3} T^{2} \)
29 \( 1 + 272 T + p^{3} T^{2} \)
31 \( 1 - 230 T + p^{3} T^{2} \)
37 \( 1 - 182 T + p^{3} T^{2} \)
41 \( 1 - 117 T + p^{3} T^{2} \)
43 \( 1 - 372 T + p^{3} T^{2} \)
47 \( 1 + 52 T + p^{3} T^{2} \)
53 \( 1 - 402 T + p^{3} T^{2} \)
59 \( 1 + 312 T + p^{3} T^{2} \)
61 \( 1 - 170 T + p^{3} T^{2} \)
67 \( 1 - 763 T + p^{3} T^{2} \)
71 \( 1 - 52 T + p^{3} T^{2} \)
73 \( 1 - 981 T + p^{3} T^{2} \)
79 \( 1 + 1054 T + p^{3} T^{2} \)
83 \( 1 - 351 T + p^{3} T^{2} \)
89 \( 1 - 799 T + p^{3} T^{2} \)
97 \( 1 + 962 T + p^{3} T^{2} \)
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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

−11.00344689706904599269617248939, −9.700381631182984239243364021501, −9.193449668517725156435888554388, −8.057255311647584339132497917752, −7.18790999839139966597681496009, −5.99859759904397990218058062484, −5.13077044465227569466073205038, −3.63143446766723374259243797106, −2.69976128651018423430588460502, −0.918429870591323551821618234954, 0.918429870591323551821618234954, 2.69976128651018423430588460502, 3.63143446766723374259243797106, 5.13077044465227569466073205038, 5.99859759904397990218058062484, 7.18790999839139966597681496009, 8.057255311647584339132497917752, 9.193449668517725156435888554388, 9.700381631182984239243364021501, 11.00344689706904599269617248939

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