Properties

Label 2-1120-1.1-c1-0-17
Degree $2$
Conductor $1120$
Sign $-1$
Analytic cond. $8.94324$
Root an. cond. $2.99052$
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  − 3-s + 5-s + 7-s − 2·9-s − 5·11-s + 5·13-s − 15-s − 5·17-s − 21-s − 8·23-s + 25-s + 5·27-s − 29-s − 2·31-s + 5·33-s + 35-s + 4·37-s − 5·39-s + 2·41-s − 4·43-s − 2·45-s − 13·47-s + 49-s + 5·51-s − 8·53-s − 5·55-s − 4·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.50·11-s + 1.38·13-s − 0.258·15-s − 1.21·17-s − 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s − 0.359·31-s + 0.870·33-s + 0.169·35-s + 0.657·37-s − 0.800·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.89·47-s + 1/7·49-s + 0.700·51-s − 1.09·53-s − 0.674·55-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1120\)    =    \(2^{5} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(8.94324\)
Root analytic conductor: \(2.99052\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1120,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−9.491948335238814850919165746875, −8.370051000148633364855371195177, −8.063517141148345392043405722751, −6.66748854828879212944424240318, −5.94481578275946266777055244953, −5.30049955232884599582486687984, −4.29722534435929022081812306925, −2.95752036175622615716558765064, −1.82911173692261649750792126223, 0, 1.82911173692261649750792126223, 2.95752036175622615716558765064, 4.29722534435929022081812306925, 5.30049955232884599582486687984, 5.94481578275946266777055244953, 6.66748854828879212944424240318, 8.063517141148345392043405722751, 8.370051000148633364855371195177, 9.491948335238814850919165746875

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