Properties

Label 2-40e2-1.1-c1-0-14
Degree $2$
Conductor $1600$
Sign $1$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s + 9-s + 2·13-s + 6·17-s − 4·19-s + 4·21-s + 6·23-s − 4·27-s − 6·29-s + 4·31-s + 2·37-s + 4·39-s + 6·41-s + 10·43-s − 6·47-s − 3·49-s + 12·51-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s + 2·63-s − 2·67-s + 12·69-s + 12·71-s − 2·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.25·23-s − 0.769·27-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 1.68·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s + 0.251·63-s − 0.244·67-s + 1.44·69-s + 1.42·71-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.876616721\)
\(L(\frac12)\) \(\approx\) \(2.876616721\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.249279365224539237566485048625, −8.550149092441660717391493732033, −7.950176007435402201982724906846, −7.33279621647751156873215628670, −6.14181160684855424693200797613, −5.25974231276140424950498810771, −4.19097847493160731891690631788, −3.32887958847832114233485788826, −2.41662881718811067600819256762, −1.26575059491370931038306222328, 1.26575059491370931038306222328, 2.41662881718811067600819256762, 3.32887958847832114233485788826, 4.19097847493160731891690631788, 5.25974231276140424950498810771, 6.14181160684855424693200797613, 7.33279621647751156873215628670, 7.950176007435402201982724906846, 8.550149092441660717391493732033, 9.249279365224539237566485048625

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