Properties

Label 2-2e7-1.1-c1-0-3
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 $1$

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: \(1\)
Selberg data: \((2,\ 128,\ (\ :1/2),\ -1)\)

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 \( 1 \)
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

−12.44338350662649173502905410996, −11.96377224112661393265117422678, −10.90944990278511493765148592647, −9.886348706897054809183790479328, −8.635939205168057076823393046651, −6.97464122003923989664965494039, −6.33050260025483473718379487161, −4.84373431553687523132526670248, −3.36730269661969347724618284285, 0, 3.36730269661969347724618284285, 4.84373431553687523132526670248, 6.33050260025483473718379487161, 6.97464122003923989664965494039, 8.635939205168057076823393046651, 9.886348706897054809183790479328, 10.90944990278511493765148592647, 11.96377224112661393265117422678, 12.44338350662649173502905410996

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