Properties

Label 2-80e2-1.1-c1-0-2
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  − 2.82·3-s + 5.00·9-s − 2.82·11-s − 6·17-s − 8.48·19-s − 5.65·27-s + 8.00·33-s + 6·41-s − 8.48·43-s − 7·49-s + 16.9·51-s + 24·57-s − 14.1·59-s + 8.48·67-s − 2·73-s + 1.00·81-s − 2.82·83-s + 18·89-s + 10·97-s − 14.1·99-s − 19.7·107-s − 18·113-s + ⋯
L(s)  = 1  − 1.63·3-s + 1.66·9-s − 0.852·11-s − 1.45·17-s − 1.94·19-s − 1.08·27-s + 1.39·33-s + 0.937·41-s − 1.29·43-s − 49-s + 2.37·51-s + 3.17·57-s − 1.84·59-s + 1.03·67-s − 0.234·73-s + 0.111·81-s − 0.310·83-s + 1.90·89-s + 1.01·97-s − 1.42·99-s − 1.91·107-s − 1.69·113-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: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3003081091\)
\(L(\frac12)\) \(\approx\) \(0.3003081091\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 8.48T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2.82T + 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
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

−7.969927599294231266677202823616, −7.06141323459836736777381614098, −6.37686032415289551578984947366, −6.11153633976116079454683525069, −5.06387741436896575651738877777, −4.69136050196791769810329408373, −3.92955226986834837112672681695, −2.59010901828974344812434161911, −1.70611701145266943750242581733, −0.30036950091384718759111670058, 0.30036950091384718759111670058, 1.70611701145266943750242581733, 2.59010901828974344812434161911, 3.92955226986834837112672681695, 4.69136050196791769810329408373, 5.06387741436896575651738877777, 6.11153633976116079454683525069, 6.37686032415289551578984947366, 7.06141323459836736777381614098, 7.969927599294231266677202823616

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