Properties

Label 2-1155-1.1-c1-0-21
Degree $2$
Conductor $1155$
Sign $1$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34·2-s − 3-s + 3.48·4-s + 5-s − 2.34·6-s − 7-s + 3.48·8-s + 9-s + 2.34·10-s − 11-s − 3.48·12-s + 3.14·13-s − 2.34·14-s − 15-s + 1.19·16-s + 6.48·17-s + 2.34·18-s + 7.17·19-s + 3.48·20-s + 21-s − 2.34·22-s + 2.19·23-s − 3.48·24-s + 25-s + 7.37·26-s − 27-s − 3.48·28-s + ⋯
L(s)  = 1  + 1.65·2-s − 0.577·3-s + 1.74·4-s + 0.447·5-s − 0.956·6-s − 0.377·7-s + 1.23·8-s + 0.333·9-s + 0.740·10-s − 0.301·11-s − 1.00·12-s + 0.872·13-s − 0.626·14-s − 0.258·15-s + 0.299·16-s + 1.57·17-s + 0.552·18-s + 1.64·19-s + 0.780·20-s + 0.218·21-s − 0.499·22-s + 0.458·23-s − 0.712·24-s + 0.200·25-s + 1.44·26-s − 0.192·27-s − 0.659·28-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.874598782\)
\(L(\frac12)\) \(\approx\) \(3.874598782\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.34T + 2T^{2} \)
13 \( 1 - 3.14T + 13T^{2} \)
17 \( 1 - 6.48T + 17T^{2} \)
19 \( 1 - 7.17T + 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 - 0.949T + 29T^{2} \)
31 \( 1 + 6.12T + 31T^{2} \)
37 \( 1 - 0.853T + 37T^{2} \)
41 \( 1 - 3.53T + 41T^{2} \)
43 \( 1 + 5.63T + 43T^{2} \)
47 \( 1 + 1.53T + 47T^{2} \)
53 \( 1 + 6.15T + 53T^{2} \)
59 \( 1 + 5.63T + 59T^{2} \)
61 \( 1 - 4.58T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 + 8.22T + 71T^{2} \)
73 \( 1 + 4.68T + 73T^{2} \)
79 \( 1 - 2.51T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 + 5.00T + 97T^{2} \)
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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

−10.01389216270181748379046785519, −9.169616839016537032560116942572, −7.74771173946090028548696667742, −6.96517124297829632728909276927, −5.97548076295552579664459818081, −5.57141400112530864868395516464, −4.81458666021509121619292655991, −3.58632569765157522957911669834, −2.99836757066342688132379713773, −1.39686093246280100302829021436, 1.39686093246280100302829021436, 2.99836757066342688132379713773, 3.58632569765157522957911669834, 4.81458666021509121619292655991, 5.57141400112530864868395516464, 5.97548076295552579664459818081, 6.96517124297829632728909276927, 7.74771173946090028548696667742, 9.169616839016537032560116942572, 10.01389216270181748379046785519

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