Properties

Label 2-40e2-1.1-c3-0-45
Degree $2$
Conductor $1600$
Sign $1$
Analytic cond. $94.4030$
Root an. cond. $9.71612$
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  + 2·3-s + 6·7-s − 23·9-s + 60·11-s + 50·13-s + 30·17-s + 40·19-s + 12·21-s + 178·23-s − 100·27-s − 166·29-s − 20·31-s + 120·33-s + 10·37-s + 100·39-s − 250·41-s − 142·43-s + 214·47-s − 307·49-s + 60·51-s + 490·53-s + 80·57-s − 800·59-s − 250·61-s − 138·63-s + 774·67-s + 356·69-s + ⋯
L(s)  = 1  + 0.384·3-s + 0.323·7-s − 0.851·9-s + 1.64·11-s + 1.06·13-s + 0.428·17-s + 0.482·19-s + 0.124·21-s + 1.61·23-s − 0.712·27-s − 1.06·29-s − 0.115·31-s + 0.633·33-s + 0.0444·37-s + 0.410·39-s − 0.952·41-s − 0.503·43-s + 0.664·47-s − 0.895·49-s + 0.164·51-s + 1.26·53-s + 0.185·57-s − 1.76·59-s − 0.524·61-s − 0.275·63-s + 1.41·67-s + 0.621·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(94.4030\)
Root analytic conductor: \(9.71612\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.087770391\)
\(L(\frac12)\) \(\approx\) \(3.087770391\)
\(L(\frac{5}{2})\) 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 \( 1 - 2 T + p^{3} T^{2} \)
7 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 - 60 T + p^{3} T^{2} \)
13 \( 1 - 50 T + p^{3} T^{2} \)
17 \( 1 - 30 T + p^{3} T^{2} \)
19 \( 1 - 40 T + p^{3} T^{2} \)
23 \( 1 - 178 T + p^{3} T^{2} \)
29 \( 1 + 166 T + p^{3} T^{2} \)
31 \( 1 + 20 T + p^{3} T^{2} \)
37 \( 1 - 10 T + p^{3} T^{2} \)
41 \( 1 + 250 T + p^{3} T^{2} \)
43 \( 1 + 142 T + p^{3} T^{2} \)
47 \( 1 - 214 T + p^{3} T^{2} \)
53 \( 1 - 490 T + p^{3} T^{2} \)
59 \( 1 + 800 T + p^{3} T^{2} \)
61 \( 1 + 250 T + p^{3} T^{2} \)
67 \( 1 - 774 T + p^{3} T^{2} \)
71 \( 1 + 100 T + p^{3} T^{2} \)
73 \( 1 - 230 T + p^{3} T^{2} \)
79 \( 1 - 1320 T + p^{3} T^{2} \)
83 \( 1 + 982 T + p^{3} T^{2} \)
89 \( 1 - 874 T + p^{3} T^{2} \)
97 \( 1 - 310 T + p^{3} T^{2} \)
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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

−9.005773544793772886216717782579, −8.439371185824711662378308710925, −7.49137676878168659106267802288, −6.61011412400803818604075800014, −5.84045279836583876648648910451, −4.92879166154754945463816000945, −3.71726364536347033845926303426, −3.22632046002378712343837240041, −1.80797382850212160399067913493, −0.884711303870432808680614870106, 0.884711303870432808680614870106, 1.80797382850212160399067913493, 3.22632046002378712343837240041, 3.71726364536347033845926303426, 4.92879166154754945463816000945, 5.84045279836583876648648910451, 6.61011412400803818604075800014, 7.49137676878168659106267802288, 8.439371185824711662378308710925, 9.005773544793772886216717782579

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