Properties

Label 2-80e2-1.1-c1-0-103
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  + 0.317·3-s − 2.89·9-s − 3.78·11-s + 1.89·17-s + 5.97·19-s − 1.87·27-s − 1.20·33-s + 6.79·41-s + 8.48·43-s − 7·49-s + 0.603·51-s + 1.89·57-s − 14.1·59-s + 16.3·67-s − 15.6·73-s + 8.10·81-s − 17.0·83-s − 4.10·89-s − 10·97-s + 10.9·99-s − 15.0·107-s − 18.7·113-s + ⋯
L(s)  = 1  + 0.183·3-s − 0.966·9-s − 1.14·11-s + 0.460·17-s + 1.37·19-s − 0.360·27-s − 0.209·33-s + 1.06·41-s + 1.29·43-s − 49-s + 0.0845·51-s + 0.251·57-s − 1.84·59-s + 1.99·67-s − 1.83·73-s + 0.900·81-s − 1.86·83-s − 0.434·89-s − 1.01·97-s + 1.10·99-s − 1.45·107-s − 1.76·113-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: \(1\)
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 \( 1 - 0.317T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3.78T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.89T + 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 6.79T + 41T^{2} \)
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 \( 1 - 16.3T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 4.10T + 89T^{2} \)
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

−7.79130280360967418904232049454, −7.14630093955698261421195242737, −6.08316556266648971830728938643, −5.53863460502976753827261157159, −4.96494250297000239524104258498, −3.93487723900185198218735131371, −2.97072875747322522976592248663, −2.60234150203029626329223388867, −1.27324707778811480902483339128, 0, 1.27324707778811480902483339128, 2.60234150203029626329223388867, 2.97072875747322522976592248663, 3.93487723900185198218735131371, 4.96494250297000239524104258498, 5.53863460502976753827261157159, 6.08316556266648971830728938643, 7.14630093955698261421195242737, 7.79130280360967418904232049454

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