Properties

Label 2-1104-1.1-c5-0-16
Degree $2$
Conductor $1104$
Sign $1$
Analytic cond. $177.063$
Root an. cond. $13.3065$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s − 13.3·5-s − 170.·7-s + 81·9-s + 90.5·11-s + 478.·13-s + 120.·15-s + 1.13e3·17-s + 1.52e3·19-s + 1.53e3·21-s − 529·23-s − 2.94e3·25-s − 729·27-s + 5.26e3·29-s − 2.69e3·31-s − 814.·33-s + 2.27e3·35-s − 1.46e4·37-s − 4.30e3·39-s + 1.70e4·41-s − 1.69e4·43-s − 1.08e3·45-s − 1.49e4·47-s + 1.21e4·49-s − 1.02e4·51-s − 2.58e4·53-s − 1.20e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.238·5-s − 1.31·7-s + 0.333·9-s + 0.225·11-s + 0.784·13-s + 0.137·15-s + 0.954·17-s + 0.966·19-s + 0.758·21-s − 0.208·23-s − 0.943·25-s − 0.192·27-s + 1.16·29-s − 0.503·31-s − 0.130·33-s + 0.313·35-s − 1.75·37-s − 0.452·39-s + 1.58·41-s − 1.39·43-s − 0.0795·45-s − 0.989·47-s + 0.725·49-s − 0.550·51-s − 1.26·53-s − 0.0538·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(177.063\)
Root analytic conductor: \(13.3065\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.147255335\)
\(L(\frac12)\) \(\approx\) \(1.147255335\)
\(L(\frac{7}{2})\) 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 \( 1 + 9T \)
23 \( 1 + 529T \)
good5 \( 1 + 13.3T + 3.12e3T^{2} \)
7 \( 1 + 170.T + 1.68e4T^{2} \)
11 \( 1 - 90.5T + 1.61e5T^{2} \)
13 \( 1 - 478.T + 3.71e5T^{2} \)
17 \( 1 - 1.13e3T + 1.41e6T^{2} \)
19 \( 1 - 1.52e3T + 2.47e6T^{2} \)
29 \( 1 - 5.26e3T + 2.05e7T^{2} \)
31 \( 1 + 2.69e3T + 2.86e7T^{2} \)
37 \( 1 + 1.46e4T + 6.93e7T^{2} \)
41 \( 1 - 1.70e4T + 1.15e8T^{2} \)
43 \( 1 + 1.69e4T + 1.47e8T^{2} \)
47 \( 1 + 1.49e4T + 2.29e8T^{2} \)
53 \( 1 + 2.58e4T + 4.18e8T^{2} \)
59 \( 1 + 3.90e4T + 7.14e8T^{2} \)
61 \( 1 - 1.53e4T + 8.44e8T^{2} \)
67 \( 1 + 1.56e4T + 1.35e9T^{2} \)
71 \( 1 - 2.30e4T + 1.80e9T^{2} \)
73 \( 1 - 8.90e3T + 2.07e9T^{2} \)
79 \( 1 - 7.27e4T + 3.07e9T^{2} \)
83 \( 1 + 7.47e4T + 3.93e9T^{2} \)
89 \( 1 - 4.23e4T + 5.58e9T^{2} \)
97 \( 1 - 1.10e5T + 8.58e9T^{2} \)
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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

−9.350324036293984230074882644301, −8.252238884164178187632576035694, −7.36318403786176921865830935809, −6.46762404002218887998890962073, −5.90698247937715046804635054264, −4.92635440199336092729912550646, −3.66579055682679222339997124224, −3.17289245462700317133946384179, −1.55823047681001657034128990646, −0.48741643721086589421368365505, 0.48741643721086589421368365505, 1.55823047681001657034128990646, 3.17289245462700317133946384179, 3.66579055682679222339997124224, 4.92635440199336092729912550646, 5.90698247937715046804635054264, 6.46762404002218887998890962073, 7.36318403786176921865830935809, 8.252238884164178187632576035694, 9.350324036293984230074882644301

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