Properties

Label 2-1104-1.1-c5-0-105
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 + 52.3·5-s + 118.·7-s + 81·9-s − 741.·11-s − 542.·13-s + 471.·15-s − 834.·17-s + 1.30e3·19-s + 1.06e3·21-s − 529·23-s − 380.·25-s + 729·27-s + 5.53e3·29-s + 7.26e3·31-s − 6.67e3·33-s + 6.19e3·35-s − 1.04e4·37-s − 4.88e3·39-s − 4.63e3·41-s + 9.20e3·43-s + 4.24e3·45-s − 1.52e4·47-s − 2.82e3·49-s − 7.51e3·51-s − 3.62e4·53-s − 3.88e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.937·5-s + 0.912·7-s + 0.333·9-s − 1.84·11-s − 0.890·13-s + 0.541·15-s − 0.700·17-s + 0.832·19-s + 0.526·21-s − 0.208·23-s − 0.121·25-s + 0.192·27-s + 1.22·29-s + 1.35·31-s − 1.06·33-s + 0.854·35-s − 1.25·37-s − 0.514·39-s − 0.430·41-s + 0.758·43-s + 0.312·45-s − 1.00·47-s − 0.168·49-s − 0.404·51-s − 1.77·53-s − 1.73·55-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 \( 1 - 52.3T + 3.12e3T^{2} \)
7 \( 1 - 118.T + 1.68e4T^{2} \)
11 \( 1 + 741.T + 1.61e5T^{2} \)
13 \( 1 + 542.T + 3.71e5T^{2} \)
17 \( 1 + 834.T + 1.41e6T^{2} \)
19 \( 1 - 1.30e3T + 2.47e6T^{2} \)
29 \( 1 - 5.53e3T + 2.05e7T^{2} \)
31 \( 1 - 7.26e3T + 2.86e7T^{2} \)
37 \( 1 + 1.04e4T + 6.93e7T^{2} \)
41 \( 1 + 4.63e3T + 1.15e8T^{2} \)
43 \( 1 - 9.20e3T + 1.47e8T^{2} \)
47 \( 1 + 1.52e4T + 2.29e8T^{2} \)
53 \( 1 + 3.62e4T + 4.18e8T^{2} \)
59 \( 1 + 6.43e3T + 7.14e8T^{2} \)
61 \( 1 + 2.63e3T + 8.44e8T^{2} \)
67 \( 1 + 2.10e4T + 1.35e9T^{2} \)
71 \( 1 + 7.46e4T + 1.80e9T^{2} \)
73 \( 1 - 8.07e4T + 2.07e9T^{2} \)
79 \( 1 + 3.91e4T + 3.07e9T^{2} \)
83 \( 1 + 1.01e5T + 3.93e9T^{2} \)
89 \( 1 - 2.24e4T + 5.58e9T^{2} \)
97 \( 1 + 2.97e4T + 8.58e9T^{2} \)
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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

−8.568404036151086185974001437118, −7.980544408115697439248236483252, −7.24626641769350138875837986838, −6.11742872838156129671536686729, −5.04594377237544225910207643949, −4.70745460999302684619329970145, −3.01113891022234666146132778831, −2.36936008608817278161409949679, −1.49095491873613620706202917302, 0, 1.49095491873613620706202917302, 2.36936008608817278161409949679, 3.01113891022234666146132778831, 4.70745460999302684619329970145, 5.04594377237544225910207643949, 6.11742872838156129671536686729, 7.24626641769350138875837986838, 7.980544408115697439248236483252, 8.568404036151086185974001437118

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