Properties

Label 2-5712-1.1-c1-0-69
Degree $2$
Conductor $5712$
Sign $-1$
Analytic cond. $45.6105$
Root an. cond. $6.75355$
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 − 2·5-s + 7-s + 9-s − 6·13-s − 2·15-s + 17-s + 21-s + 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 2·35-s + 10·37-s − 6·39-s − 6·41-s − 12·43-s − 2·45-s + 49-s + 51-s − 10·53-s + 8·59-s + 6·61-s + 63-s + 12·65-s − 12·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s + 0.242·17-s + 0.218·21-s + 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.338·35-s + 1.64·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s − 0.298·45-s + 1/7·49-s + 0.140·51-s − 1.37·53-s + 1.04·59-s + 0.768·61-s + 0.125·63-s + 1.48·65-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5712\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(45.6105\)
Root analytic conductor: \(6.75355\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5712,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
17 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
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

−7.87926904976884309319145553824, −7.19515866090246264939530945112, −6.63441653407793169712897922103, −5.37292810110458713350651257174, −4.77235890347525183836188577110, −4.11402402903198192748250030289, −3.14603790489072703936478308725, −2.53767352748806757689866707676, −1.36698861038099470242315828550, 0, 1.36698861038099470242315828550, 2.53767352748806757689866707676, 3.14603790489072703936478308725, 4.11402402903198192748250030289, 4.77235890347525183836188577110, 5.37292810110458713350651257174, 6.63441653407793169712897922103, 7.19515866090246264939530945112, 7.87926904976884309319145553824

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