Properties

Label 2-40e2-1.1-c1-0-11
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  + 3-s + 2·7-s − 2·9-s + 5·11-s + 5·17-s + 5·19-s + 2·21-s − 6·23-s − 5·27-s − 4·29-s − 10·31-s + 5·33-s + 10·37-s + 5·41-s + 4·43-s + 8·47-s − 3·49-s + 5·51-s + 10·53-s + 5·57-s + 10·61-s − 4·63-s + 3·67-s − 6·69-s − 5·73-s + 10·77-s − 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s + 1.50·11-s + 1.21·17-s + 1.14·19-s + 0.436·21-s − 1.25·23-s − 0.962·27-s − 0.742·29-s − 1.79·31-s + 0.870·33-s + 1.64·37-s + 0.780·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.700·51-s + 1.37·53-s + 0.662·57-s + 1.28·61-s − 0.503·63-s + 0.366·67-s − 0.722·69-s − 0.585·73-s + 1.13·77-s − 1.12·79-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

\(L(1)\) \(\approx\) \(2.420509571\)
\(L(\frac12)\) \(\approx\) \(2.420509571\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−9.323754252560504956925848056540, −8.683366516272630610807238108491, −7.76550095341300361088846412304, −7.32233528761629317808083018374, −5.94672002738926315494959415182, −5.49835151922875749387532575327, −4.11708760011461050263971856342, −3.52621022238586572141128158530, −2.29156961942451897610628689947, −1.16753716752208245348849419893, 1.16753716752208245348849419893, 2.29156961942451897610628689947, 3.52621022238586572141128158530, 4.11708760011461050263971856342, 5.49835151922875749387532575327, 5.94672002738926315494959415182, 7.32233528761629317808083018374, 7.76550095341300361088846412304, 8.683366516272630610807238108491, 9.323754252560504956925848056540

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