Properties

Label 2-72e2-1.1-c1-0-62
Degree $2$
Conductor $5184$
Sign $-1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-1$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5184} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ -1)\)

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

−7.979366646321461123720553135432, −7.06966982994878128889525387936, −6.23645271216033822133115751353, −5.88323182271317659077662603339, −5.01147268838399809080141393398, −4.06818689919942569274238278977, −3.10804393408213309934618532197, −2.60696863261941228826025935794, −1.31755823105286963939730215101, 0, 1.31755823105286963939730215101, 2.60696863261941228826025935794, 3.10804393408213309934618532197, 4.06818689919942569274238278977, 5.01147268838399809080141393398, 5.88323182271317659077662603339, 6.23645271216033822133115751353, 7.06966982994878128889525387936, 7.979366646321461123720553135432

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