Properties

Label 2-80e2-1.1-c1-0-30
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  + 2.82·7-s − 3·9-s − 6.32·11-s + 4.47·13-s − 6.32·19-s + 8.48·23-s − 4.47·37-s + 2·41-s − 2.82·47-s + 1.00·49-s + 13.4·53-s + 6.32·59-s − 8.48·63-s − 17.8·77-s + 9·81-s + 14·89-s + 12.6·91-s + 18.9·99-s + 19.7·103-s − 13.4·117-s + ⋯
L(s)  = 1  + 1.06·7-s − 9-s − 1.90·11-s + 1.24·13-s − 1.45·19-s + 1.76·23-s − 0.735·37-s + 0.312·41-s − 0.412·47-s + 0.142·49-s + 1.84·53-s + 0.823·59-s − 1.06·63-s − 2.03·77-s + 81-s + 1.48·89-s + 1.32·91-s + 1.90·99-s + 1.95·103-s − 1.24·117-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

\(L(1)\) \(\approx\) \(1.717792583\)
\(L(\frac12)\) \(\approx\) \(1.717792583\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 6.32T + 11T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 6.32T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.194053751078541039485379451500, −7.46653989479500886474975013995, −6.59524664052081623388234805279, −5.72690018408399215610948321705, −5.21616069665935701661215643404, −4.59507693393422520929209621113, −3.52413751406553569516076847105, −2.68938403856704086596122066369, −1.95705241053648723725728959061, −0.66484135115214633738124175840, 0.66484135115214633738124175840, 1.95705241053648723725728959061, 2.68938403856704086596122066369, 3.52413751406553569516076847105, 4.59507693393422520929209621113, 5.21616069665935701661215643404, 5.72690018408399215610948321705, 6.59524664052081623388234805279, 7.46653989479500886474975013995, 8.194053751078541039485379451500

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