Properties

Label 2-6776-1.1-c1-0-136
Degree $2$
Conductor $6776$
Sign $-1$
Analytic cond. $54.1066$
Root an. cond. $7.35572$
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  + 2·5-s + 7-s − 3·9-s − 2·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s + 4·43-s − 6·45-s − 8·47-s + 49-s + 6·53-s + 6·61-s − 3·63-s − 4·65-s − 4·67-s − 8·71-s − 10·73-s − 16·79-s + 9·81-s − 8·83-s + 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s − 9-s − 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.768·61-s − 0.377·63-s − 0.496·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s − 1.80·79-s + 81-s − 0.878·83-s + 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6776\)    =    \(2^{3} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(54.1066\)
Root analytic conductor: \(7.35572\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6776} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6776,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.69587064441186150768078543731, −6.88310002274215763839569601337, −5.94974241082854154889853475745, −5.72315905059004231786170242312, −4.86884684581528043387154729760, −4.03151585237256402081332864489, −2.98780927444278559643957768594, −2.30074942792092931708066021417, −1.44245425010921288807179765849, 0, 1.44245425010921288807179765849, 2.30074942792092931708066021417, 2.98780927444278559643957768594, 4.03151585237256402081332864489, 4.86884684581528043387154729760, 5.72315905059004231786170242312, 5.94974241082854154889853475745, 6.88310002274215763839569601337, 7.69587064441186150768078543731

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