Properties

Label 2-80e2-1.1-c1-0-73
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  − 1.41·3-s − 4.24·7-s − 0.999·9-s + 5.65·11-s + 2·13-s − 6·17-s − 2.82·19-s + 6·21-s + 7.07·23-s + 5.65·27-s − 4·29-s − 2.82·31-s − 8.00·33-s − 2·37-s − 2.82·39-s + 8·41-s + 1.41·43-s + 1.41·47-s + 10.9·49-s + 8.48·51-s + 2·53-s + 4.00·57-s + 2.82·59-s − 14·61-s + 4.24·63-s + 4.24·67-s − 10.0·69-s + ⋯
L(s)  = 1  − 0.816·3-s − 1.60·7-s − 0.333·9-s + 1.70·11-s + 0.554·13-s − 1.45·17-s − 0.648·19-s + 1.30·21-s + 1.47·23-s + 1.08·27-s − 0.742·29-s − 0.508·31-s − 1.39·33-s − 0.328·37-s − 0.452·39-s + 1.24·41-s + 0.215·43-s + 0.206·47-s + 1.57·49-s + 1.18·51-s + 0.274·53-s + 0.529·57-s + 0.368·59-s − 1.79·61-s + 0.534·63-s + 0.518·67-s − 1.20·69-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 \( 1 + 1.41T + 3T^{2} \)
7 \( 1 + 4.24T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + 2.82T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 1.41T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4.24T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.36528255984158725879437767433, −6.64099580238875886902044522286, −6.36624038918159468167990757880, −5.83091907376434779232866008512, −4.79044364675678882530820483590, −3.95563457272609149645435562973, −3.35097176825828344812189464820, −2.34632534166507935123728574134, −1.03410111338269704050176092180, 0, 1.03410111338269704050176092180, 2.34632534166507935123728574134, 3.35097176825828344812189464820, 3.95563457272609149645435562973, 4.79044364675678882530820483590, 5.83091907376434779232866008512, 6.36624038918159468167990757880, 6.64099580238875886902044522286, 7.36528255984158725879437767433

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