Properties

Label 2-2e7-1.1-c1-0-2
Degree $2$
Conductor $128$
Sign $1$
Analytic cond. $1.02208$
Root an. cond. $1.01098$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s − 4·7-s + 9-s − 2·11-s + 2·13-s + 4·15-s − 2·17-s + 2·19-s − 8·21-s + 4·23-s − 25-s − 4·27-s − 6·29-s − 4·33-s − 8·35-s + 10·37-s + 4·39-s − 6·41-s + 6·43-s + 2·45-s − 8·47-s + 9·49-s − 4·51-s − 6·53-s − 4·55-s + 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.485·17-s + 0.458·19-s − 1.74·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s − 1.11·29-s − 0.696·33-s − 1.35·35-s + 1.64·37-s + 0.640·39-s − 0.937·41-s + 0.914·43-s + 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.560·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $1$
Analytic conductor: \(1.02208\)
Root analytic conductor: \(1.01098\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427104703\)
\(L(\frac12)\) \(\approx\) \(1.427104703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.19000280536280595734718108817, −13.01019465812383259987837132361, −11.15482415732823435746426270865, −9.742591354048855107896325616652, −9.385355573475711088007535446099, −8.173183686627627779120693205031, −6.78571919296342801026858814862, −5.64130249043000575312616744005, −3.56276783709589056726776122484, −2.46608091328452001690059553408, 2.46608091328452001690059553408, 3.56276783709589056726776122484, 5.64130249043000575312616744005, 6.78571919296342801026858814862, 8.173183686627627779120693205031, 9.385355573475711088007535446099, 9.742591354048855107896325616652, 11.15482415732823435746426270865, 13.01019465812383259987837132361, 13.19000280536280595734718108817

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