Properties

Label 2-80e2-1.1-c1-0-46
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  − 2.44·3-s + 1.41·7-s + 2.99·9-s + 2·11-s + 5.65·13-s − 4.89·17-s + 6·19-s − 3.46·21-s + 7.07·23-s + 6.92·29-s + 6.92·31-s − 4.89·33-s − 2.82·37-s − 13.8·39-s + 4·41-s + 2.44·43-s − 4.24·47-s − 5·49-s + 11.9·51-s − 14.6·57-s + 2·59-s + 3.46·61-s + 4.24·63-s − 2.44·67-s − 17.3·69-s + 6.92·71-s − 4.89·73-s + ⋯
L(s)  = 1  − 1.41·3-s + 0.534·7-s + 0.999·9-s + 0.603·11-s + 1.56·13-s − 1.18·17-s + 1.37·19-s − 0.755·21-s + 1.47·23-s + 1.28·29-s + 1.24·31-s − 0.852·33-s − 0.464·37-s − 2.21·39-s + 0.624·41-s + 0.373·43-s − 0.618·47-s − 0.714·49-s + 1.68·51-s − 1.94·57-s + 0.260·59-s + 0.443·61-s + 0.534·63-s − 0.299·67-s − 2.08·69-s + 0.822·71-s − 0.573·73-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

\(L(1)\) \(\approx\) \(1.645523147\)
\(L(\frac12)\) \(\approx\) \(1.645523147\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 2.44T + 43T^{2} \)
47 \( 1 + 4.24T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 3.46T + 61T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 - 6.92T + 71T^{2} \)
73 \( 1 + 4.89T + 73T^{2} \)
79 \( 1 - 6.92T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−8.061462458852688913675420180096, −6.99207337006568627001244575902, −6.53031462836510165127397855344, −5.99957053827793767549068662289, −5.09383578230924223767749892298, −4.70295489216631238853998080135, −3.78441862009535141334696670362, −2.79751932829467436319646808086, −1.35265479541562967340386683530, −0.845458985825921989627349439376, 0.845458985825921989627349439376, 1.35265479541562967340386683530, 2.79751932829467436319646808086, 3.78441862009535141334696670362, 4.70295489216631238853998080135, 5.09383578230924223767749892298, 5.99957053827793767549068662289, 6.53031462836510165127397855344, 6.99207337006568627001244575902, 8.061462458852688913675420180096

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