Properties

Label 2-80e2-1.1-c1-0-142
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  + 3.14·3-s + 6.89·9-s − 6.61·11-s − 7.89·17-s − 2.51·19-s + 12.2·27-s − 20.7·33-s − 12.7·41-s − 8.48·43-s − 7·49-s − 24.8·51-s − 7.89·57-s + 14.1·59-s + 7.88·67-s + 13.6·73-s + 17.8·81-s − 14.1·83-s − 13.8·89-s − 10·97-s − 45.6·99-s + 4.70·107-s + 0.797·113-s + ⋯
L(s)  = 1  + 1.81·3-s + 2.29·9-s − 1.99·11-s − 1.91·17-s − 0.575·19-s + 2.36·27-s − 3.62·33-s − 1.99·41-s − 1.29·43-s − 49-s − 3.48·51-s − 1.04·57-s + 1.84·59-s + 0.962·67-s + 1.60·73-s + 1.98·81-s − 1.55·83-s − 1.47·89-s − 1.01·97-s − 4.58·99-s + 0.454·107-s + 0.0750·113-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 \( 1 - 3.14T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 6.61T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 7.89T + 17T^{2} \)
19 \( 1 + 2.51T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
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 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 13.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
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

−7.981879129366267325931321713326, −7.03966508486081567698353855764, −6.65241609387333316709260352563, −5.29344322942415186543109693780, −4.67489823635843136447572106622, −3.85275226924941843631452294886, −3.02092072988580322889793501959, −2.37525137814252468389312573165, −1.83755944889717151008099956773, 0, 1.83755944889717151008099956773, 2.37525137814252468389312573165, 3.02092072988580322889793501959, 3.85275226924941843631452294886, 4.67489823635843136447572106622, 5.29344322942415186543109693780, 6.65241609387333316709260352563, 7.03966508486081567698353855764, 7.981879129366267325931321713326

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