Properties

Label 2-77-1.1-c1-0-4
Degree $2$
Conductor $77$
Sign $-1$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 2·4-s − 5-s − 7-s + 6·9-s − 11-s + 6·12-s − 4·13-s + 3·15-s + 4·16-s + 2·17-s − 6·19-s + 2·20-s + 3·21-s − 5·23-s − 4·25-s − 9·27-s + 2·28-s + 10·29-s + 31-s + 3·33-s + 35-s − 12·36-s − 5·37-s + 12·39-s − 2·41-s − 8·43-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s − 0.447·5-s − 0.377·7-s + 2·9-s − 0.301·11-s + 1.73·12-s − 1.10·13-s + 0.774·15-s + 16-s + 0.485·17-s − 1.37·19-s + 0.447·20-s + 0.654·21-s − 1.04·23-s − 4/5·25-s − 1.73·27-s + 0.377·28-s + 1.85·29-s + 0.179·31-s + 0.522·33-s + 0.169·35-s − 2·36-s − 0.821·37-s + 1.92·39-s − 0.312·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -1)\)

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)$
bad7 \( 1 + T \)
11 \( 1 + T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.75798057830490060611386820011, −12.41538070383590792560753216926, −12.09579448828259015201597448613, −10.54462273276049844454718422984, −9.842751379499007423618867754387, −8.090125574816450530034167252480, −6.56709274805536909588701134945, −5.30170245060560361736841457140, −4.26020787258843561459488122037, 0, 4.26020787258843561459488122037, 5.30170245060560361736841457140, 6.56709274805536909588701134945, 8.090125574816450530034167252480, 9.842751379499007423618867754387, 10.54462273276049844454718422984, 12.09579448828259015201597448613, 12.41538070383590792560753216926, 13.75798057830490060611386820011

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