Properties

Label 2-648-1.1-c1-0-8
Degree $2$
Conductor $648$
Sign $-1$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  − 5-s − 3·7-s + 5·11-s − 5·13-s − 2·17-s − 4·19-s − 23-s − 4·25-s − 9·29-s − 31-s + 3·35-s − 6·37-s + 3·41-s + 43-s − 3·47-s + 2·49-s + 2·53-s − 5·55-s + 11·59-s + 7·61-s + 5·65-s − 67-s + 4·71-s − 2·73-s − 15·77-s + 79-s + 83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 1.50·11-s − 1.38·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s − 4/5·25-s − 1.67·29-s − 0.179·31-s + 0.507·35-s − 0.986·37-s + 0.468·41-s + 0.152·43-s − 0.437·47-s + 2/7·49-s + 0.274·53-s − 0.674·55-s + 1.43·59-s + 0.896·61-s + 0.620·65-s − 0.122·67-s + 0.474·71-s − 0.234·73-s − 1.70·77-s + 0.112·79-s + 0.109·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{648} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−9.872166086054390182131680931495, −9.425062765356892372201588625238, −8.488693062062992053262410459368, −7.24964163655286011443852633632, −6.70196192600962287834875283080, −5.67914881339550181972392218784, −4.28403219298754109102003210043, −3.56271769221553523884362414547, −2.12247216163999118504559644758, 0, 2.12247216163999118504559644758, 3.56271769221553523884362414547, 4.28403219298754109102003210043, 5.67914881339550181972392218784, 6.70196192600962287834875283080, 7.24964163655286011443852633632, 8.488693062062992053262410459368, 9.425062765356892372201588625238, 9.872166086054390182131680931495

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