Properties

Label 2-80e2-1.1-c1-0-107
Degree $2$
Conductor $6400$
Sign $-1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732·3-s − 2.73·7-s − 2.46·9-s + 2·11-s − 3.46·13-s + 3.46·17-s + 7.46·19-s − 2·21-s + 4.19·23-s − 4·27-s − 6.92·29-s + 1.46·31-s + 1.46·33-s − 2·37-s − 2.53·39-s + 5.46·41-s − 8.73·43-s − 6.73·47-s + 0.464·49-s + 2.53·51-s + 4.53·53-s + 5.46·57-s + 0.535·59-s + 4.92·61-s + 6.73·63-s − 7.26·67-s + 3.07·69-s + ⋯
L(s)  = 1  + 0.422·3-s − 1.03·7-s − 0.821·9-s + 0.603·11-s − 0.960·13-s + 0.840·17-s + 1.71·19-s − 0.436·21-s + 0.874·23-s − 0.769·27-s − 1.28·29-s + 0.262·31-s + 0.254·33-s − 0.328·37-s − 0.406·39-s + 0.853·41-s − 1.33·43-s − 0.981·47-s + 0.0663·49-s + 0.355·51-s + 0.623·53-s + 0.723·57-s + 0.0697·59-s + 0.630·61-s + 0.848·63-s − 0.887·67-s + 0.369·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(51.1042\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732T + 3T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 7.46T + 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 5.46T + 41T^{2} \)
43 \( 1 + 8.73T + 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 - 4.53T + 53T^{2} \)
59 \( 1 - 0.535T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 7.26T + 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 - 4.92T + 89T^{2} \)
97 \( 1 + 6.39T + 97T^{2} \)
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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

−7.51267124893288286738546843904, −7.16155534845415713616108572187, −6.24602930633120868417973549712, −5.53212893275684604920367679819, −4.93938191233125999326760709179, −3.67018408792319571140739612666, −3.23350412118007930517527162137, −2.55296280200935095951953355786, −1.28948747315570848740068437462, 0, 1.28948747315570848740068437462, 2.55296280200935095951953355786, 3.23350412118007930517527162137, 3.67018408792319571140739612666, 4.93938191233125999326760709179, 5.53212893275684604920367679819, 6.24602930633120868417973549712, 7.16155534845415713616108572187, 7.51267124893288286738546843904

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