Properties

Label 2-80e2-1.1-c1-0-124
Degree $2$
Conductor $6400$
Sign $-1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s − 6·11-s + 6·17-s − 2·19-s − 4·27-s − 12·33-s + 6·41-s − 10·43-s − 7·49-s + 12·51-s − 4·57-s − 6·59-s − 14·67-s + 2·73-s − 11·81-s + 18·83-s − 18·89-s − 10·97-s − 6·99-s + 6·107-s − 18·113-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s − 1.80·11-s + 1.45·17-s − 0.458·19-s − 0.769·27-s − 2.08·33-s + 0.937·41-s − 1.52·43-s − 49-s + 1.68·51-s − 0.529·57-s − 0.781·59-s − 1.71·67-s + 0.234·73-s − 1.22·81-s + 1.97·83-s − 1.90·89-s − 1.01·97-s − 0.603·99-s + 0.580·107-s − 1.69·113-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: yes
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.916065585735103283652712175009, −7.30234586820208772911358973986, −6.22659970710716983540786319408, −5.45696413973074733628294542781, −4.83922622654215378860083695038, −3.77827710629714312463956266694, −3.02788078297614079450569073043, −2.55749658671208684936297811870, −1.54300183056550727297791933372, 0, 1.54300183056550727297791933372, 2.55749658671208684936297811870, 3.02788078297614079450569073043, 3.77827710629714312463956266694, 4.83922622654215378860083695038, 5.45696413973074733628294542781, 6.22659970710716983540786319408, 7.30234586820208772911358973986, 7.916065585735103283652712175009

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