Properties

Label 2-80e2-1.1-c1-0-19
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  − 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: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.167457700\)
\(L(\frac12)\) \(\approx\) \(1.167457700\)
\(L(\frac{3}{2})\) 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

−7.88578303990572698048136053157, −7.52702765074133946574065599794, −6.50297766202241028702288953715, −5.87575390520710166741641744260, −5.15589358326795781860947298375, −4.64938002474910258107931848253, −3.37096668253640155189772637869, −2.89033203408306651148939339900, −1.89592775558909273271737096266, −0.54518757079496570592012833740, 0.54518757079496570592012833740, 1.89592775558909273271737096266, 2.89033203408306651148939339900, 3.37096668253640155189772637869, 4.64938002474910258107931848253, 5.15589358326795781860947298375, 5.87575390520710166741641744260, 6.50297766202241028702288953715, 7.52702765074133946574065599794, 7.88578303990572698048136053157

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