Properties

Label 2-1104-1.1-c1-0-12
Degree $2$
Conductor $1104$
Sign $1$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4·5-s − 2·7-s + 9-s + 2·13-s + 4·15-s − 4·17-s + 6·19-s − 2·21-s − 23-s + 11·25-s + 27-s + 10·29-s − 4·31-s − 8·35-s − 2·37-s + 2·39-s − 6·41-s + 6·43-s + 4·45-s − 8·47-s − 3·49-s − 4·51-s + 8·53-s + 6·57-s + 4·59-s − 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.03·15-s − 0.970·17-s + 1.37·19-s − 0.436·21-s − 0.208·23-s + 11/5·25-s + 0.192·27-s + 1.85·29-s − 0.718·31-s − 1.35·35-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 0.914·43-s + 0.596·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s + 1.09·53-s + 0.794·57-s + 0.520·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.604473046\)
\(L(\frac12)\) \(\approx\) \(2.604473046\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−9.914006807676669962241571423348, −9.070041446505792625715217792931, −8.542943637336153365831417733496, −7.16621209719492293549597219375, −6.45124423306481058213156835335, −5.73344871773213011915284599901, −4.74193564675495944765137153966, −3.33397567420318910479873505267, −2.50126630850099212592029473116, −1.38662067170928641980205842806, 1.38662067170928641980205842806, 2.50126630850099212592029473116, 3.33397567420318910479873505267, 4.74193564675495944765137153966, 5.73344871773213011915284599901, 6.45124423306481058213156835335, 7.16621209719492293549597219375, 8.542943637336153365831417733496, 9.070041446505792625715217792931, 9.914006807676669962241571423348

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