Properties

Label 2-1155-1.1-c1-0-36
Degree $2$
Conductor $1155$
Sign $-1$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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-s − 3-s − 4-s + 5-s − 6-s + 7-s − 3·8-s + 9-s + 10-s − 11-s + 12-s − 2·13-s + 14-s − 15-s − 16-s − 6·17-s + 18-s + 4·19-s − 20-s − 21-s − 22-s − 4·23-s + 3·24-s + 25-s − 2·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.438199599986511453147505381379, −8.688876376173663543847229264643, −7.63560441370956662338729532790, −6.62629853665671709769618392561, −5.77641410636928002092460840171, −5.03185863253334296879036362235, −4.42103096904459177558971112803, −3.24231752509856341716457202217, −1.90210262429212048684962532791, 0, 1.90210262429212048684962532791, 3.24231752509856341716457202217, 4.42103096904459177558971112803, 5.03185863253334296879036362235, 5.77641410636928002092460840171, 6.62629853665671709769618392561, 7.63560441370956662338729532790, 8.688876376173663543847229264643, 9.438199599986511453147505381379

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