Properties

Label 2-1155-1.1-c1-0-35
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  + 2.44·2-s + 3-s + 3.99·4-s + 5-s + 2.44·6-s + 7-s + 4.89·8-s + 9-s + 2.44·10-s − 11-s + 3.99·12-s − 0.449·13-s + 2.44·14-s + 15-s + 3.99·16-s − 3·17-s + 2.44·18-s − 19-s + 3.99·20-s + 21-s − 2.44·22-s − 1.89·23-s + 4.89·24-s + 25-s − 1.10·26-s + 27-s + 3.99·28-s + ⋯
L(s)  = 1  + 1.73·2-s + 0.577·3-s + 1.99·4-s + 0.447·5-s + 0.999·6-s + 0.377·7-s + 1.73·8-s + 0.333·9-s + 0.774·10-s − 0.301·11-s + 1.15·12-s − 0.124·13-s + 0.654·14-s + 0.258·15-s + 0.999·16-s − 0.727·17-s + 0.577·18-s − 0.229·19-s + 0.894·20-s + 0.218·21-s − 0.522·22-s − 0.395·23-s + 0.999·24-s + 0.200·25-s − 0.215·26-s + 0.192·27-s + 0.755·28-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

\(L(1)\) \(\approx\) \(5.703176951\)
\(L(\frac12)\) \(\approx\) \(5.703176951\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 2T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 + 0.550T + 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 - 4.44T + 37T^{2} \)
41 \( 1 - 2.44T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 + 8.44T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 - 0.550T + 59T^{2} \)
61 \( 1 + 2.10T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 3.10T + 73T^{2} \)
79 \( 1 - 3.34T + 79T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 3.44T + 97T^{2} \)
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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

−9.909423917098188679026140748774, −8.953093057634432748978205323460, −7.937880742304264481796786981239, −7.04943385573894750509215927666, −6.22703458914111482835758646618, −5.36943391136335975818755921160, −4.56227699011763043193954203653, −3.74553808317078905728907180294, −2.66938400307806483527995700217, −1.89371464819143966480518331598, 1.89371464819143966480518331598, 2.66938400307806483527995700217, 3.74553808317078905728907180294, 4.56227699011763043193954203653, 5.36943391136335975818755921160, 6.22703458914111482835758646618, 7.04943385573894750509215927666, 7.937880742304264481796786981239, 8.953093057634432748978205323460, 9.909423917098188679026140748774

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