Properties

Label 2-80e2-1.1-c1-0-68
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.46·7-s − 2·9-s + 3·11-s + 3.46·13-s + 3·17-s − 19-s + 3.46·21-s − 5·27-s + 10.3·29-s + 6.92·31-s + 3·33-s − 10.3·37-s + 3.46·39-s + 9·41-s + 4·43-s − 10.3·47-s + 4.99·49-s + 3·51-s − 57-s − 12·59-s − 3.46·61-s − 6.92·63-s + 11·67-s − 10.3·71-s − 7·73-s + 10.3·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.30·7-s − 0.666·9-s + 0.904·11-s + 0.960·13-s + 0.727·17-s − 0.229·19-s + 0.755·21-s − 0.962·27-s + 1.92·29-s + 1.24·31-s + 0.522·33-s − 1.70·37-s + 0.554·39-s + 1.40·41-s + 0.609·43-s − 1.51·47-s + 0.714·49-s + 0.420·51-s − 0.132·57-s − 1.56·59-s − 0.443·61-s − 0.872·63-s + 1.34·67-s − 1.23·71-s − 0.819·73-s + 1.18·77-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.355478967\)
\(L(\frac12)\) \(\approx\) \(3.355478967\)
\(L(\frac{3}{2})\) 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 \( 1 - T + 3T^{2} \)
7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 3.46T + 61T^{2} \)
67 \( 1 - 11T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.092215483860948624802294134786, −7.60334690251796714033122268217, −6.48762318905448830540571611019, −6.03131544470245267037028427810, −5.03995983879745872141396852651, −4.42917273509330128606505983426, −3.53982147367611561777893790859, −2.82141262863596428801118183360, −1.75753263798141382406855779200, −1.01051541151638899632887973399, 1.01051541151638899632887973399, 1.75753263798141382406855779200, 2.82141262863596428801118183360, 3.53982147367611561777893790859, 4.42917273509330128606505983426, 5.03995983879745872141396852651, 6.03131544470245267037028427810, 6.48762318905448830540571611019, 7.60334690251796714033122268217, 8.092215483860948624802294134786

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