Properties

Label 2-1104-1.1-c5-0-104
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 64.5·5-s + 11.9·7-s + 81·9-s − 303.·11-s − 771.·13-s + 580.·15-s + 401.·17-s + 273.·19-s + 107.·21-s + 529·23-s + 1.03e3·25-s + 729·27-s − 5.60e3·29-s + 1.95e3·31-s − 2.72e3·33-s + 773.·35-s − 6.65e3·37-s − 6.94e3·39-s + 1.35e4·41-s + 2.08e4·43-s + 5.22e3·45-s − 2.76e4·47-s − 1.66e4·49-s + 3.61e3·51-s + 1.62e4·53-s − 1.95e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.15·5-s + 0.0925·7-s + 0.333·9-s − 0.755·11-s − 1.26·13-s + 0.666·15-s + 0.337·17-s + 0.173·19-s + 0.0534·21-s + 0.208·23-s + 0.331·25-s + 0.192·27-s − 1.23·29-s + 0.365·31-s − 0.436·33-s + 0.106·35-s − 0.799·37-s − 0.730·39-s + 1.26·41-s + 1.72·43-s + 0.384·45-s − 1.82·47-s − 0.991·49-s + 0.194·51-s + 0.796·53-s − 0.871·55-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: \(1\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 64.5T + 3.12e3T^{2} \)
7 \( 1 - 11.9T + 1.68e4T^{2} \)
11 \( 1 + 303.T + 1.61e5T^{2} \)
13 \( 1 + 771.T + 3.71e5T^{2} \)
17 \( 1 - 401.T + 1.41e6T^{2} \)
19 \( 1 - 273.T + 2.47e6T^{2} \)
29 \( 1 + 5.60e3T + 2.05e7T^{2} \)
31 \( 1 - 1.95e3T + 2.86e7T^{2} \)
37 \( 1 + 6.65e3T + 6.93e7T^{2} \)
41 \( 1 - 1.35e4T + 1.15e8T^{2} \)
43 \( 1 - 2.08e4T + 1.47e8T^{2} \)
47 \( 1 + 2.76e4T + 2.29e8T^{2} \)
53 \( 1 - 1.62e4T + 4.18e8T^{2} \)
59 \( 1 + 1.17e4T + 7.14e8T^{2} \)
61 \( 1 + 2.03e4T + 8.44e8T^{2} \)
67 \( 1 + 4.96e4T + 1.35e9T^{2} \)
71 \( 1 + 1.86e3T + 1.80e9T^{2} \)
73 \( 1 + 7.51e4T + 2.07e9T^{2} \)
79 \( 1 + 4.77e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e4T + 3.93e9T^{2} \)
89 \( 1 + 8.53e4T + 5.58e9T^{2} \)
97 \( 1 + 2.59e4T + 8.58e9T^{2} \)
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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

−8.882793339101178273027953299021, −7.76667911768378989815979858920, −7.26191260520026033913321052235, −6.05674790204473024430746462868, −5.33516258584607300208641971483, −4.45499840717928409262856213269, −3.08290593860206336367915532526, −2.34817781075865006948300196469, −1.48594867335531457030862585710, 0, 1.48594867335531457030862585710, 2.34817781075865006948300196469, 3.08290593860206336367915532526, 4.45499840717928409262856213269, 5.33516258584607300208641971483, 6.05674790204473024430746462868, 7.26191260520026033913321052235, 7.76667911768378989815979858920, 8.882793339101178273027953299021

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