Properties

Label 2-1155-1.1-c1-0-34
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  − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 7-s + 9-s − 2·10-s + 11-s + 2·12-s − 6·13-s − 2·14-s + 15-s − 4·16-s − 7·17-s − 2·18-s − 5·19-s + 2·20-s + 21-s − 2·22-s − 23-s + 25-s + 12·26-s + 27-s + 2·28-s − 5·29-s − 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.577·12-s − 1.66·13-s − 0.534·14-s + 0.258·15-s − 16-s − 1.69·17-s − 0.471·18-s − 1.14·19-s + 0.447·20-s + 0.218·21-s − 0.426·22-s − 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.192·27-s + 0.377·28-s − 0.928·29-s − 0.365·30-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 \( 1 + p T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−9.212018498778742248955877369839, −8.846741941824141489575810143074, −7.87055662804250090687136539993, −7.22111789896158412118454288803, −6.44495083083266857141010156107, −5.00030635265137986421053214439, −4.12790788870104437563686789164, −2.36959749735691481963089848636, −1.87472727289654385991039771085, 0, 1.87472727289654385991039771085, 2.36959749735691481963089848636, 4.12790788870104437563686789164, 5.00030635265137986421053214439, 6.44495083083266857141010156107, 7.22111789896158412118454288803, 7.87055662804250090687136539993, 8.846741941824141489575810143074, 9.212018498778742248955877369839

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