Properties

Label 2-1104-1.1-c5-0-14
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 − 55.5·5-s − 2.50·7-s + 81·9-s − 228.·11-s + 658.·13-s + 499.·15-s − 1.44e3·17-s + 982.·19-s + 22.5·21-s − 529·23-s − 42.8·25-s − 729·27-s − 7.15e3·29-s + 9.25e3·31-s + 2.05e3·33-s + 139.·35-s + 2.42e3·37-s − 5.92e3·39-s − 4.07e3·41-s + 1.04e4·43-s − 4.49e3·45-s − 9.35e3·47-s − 1.68e4·49-s + 1.29e4·51-s − 3.42e4·53-s + 1.26e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.993·5-s − 0.0193·7-s + 0.333·9-s − 0.569·11-s + 1.08·13-s + 0.573·15-s − 1.21·17-s + 0.624·19-s + 0.0111·21-s − 0.208·23-s − 0.0137·25-s − 0.192·27-s − 1.58·29-s + 1.73·31-s + 0.328·33-s + 0.0191·35-s + 0.290·37-s − 0.624·39-s − 0.378·41-s + 0.859·43-s − 0.331·45-s − 0.617·47-s − 0.999·49-s + 0.699·51-s − 1.67·53-s + 0.565·55-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: $\chi_{1104} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7515749055\)
\(L(\frac12)\) \(\approx\) \(0.7515749055\)
\(L(\frac{7}{2})\) 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 \( 1 + 55.5T + 3.12e3T^{2} \)
7 \( 1 + 2.50T + 1.68e4T^{2} \)
11 \( 1 + 228.T + 1.61e5T^{2} \)
13 \( 1 - 658.T + 3.71e5T^{2} \)
17 \( 1 + 1.44e3T + 1.41e6T^{2} \)
19 \( 1 - 982.T + 2.47e6T^{2} \)
29 \( 1 + 7.15e3T + 2.05e7T^{2} \)
31 \( 1 - 9.25e3T + 2.86e7T^{2} \)
37 \( 1 - 2.42e3T + 6.93e7T^{2} \)
41 \( 1 + 4.07e3T + 1.15e8T^{2} \)
43 \( 1 - 1.04e4T + 1.47e8T^{2} \)
47 \( 1 + 9.35e3T + 2.29e8T^{2} \)
53 \( 1 + 3.42e4T + 4.18e8T^{2} \)
59 \( 1 - 7.26e3T + 7.14e8T^{2} \)
61 \( 1 - 2.66e4T + 8.44e8T^{2} \)
67 \( 1 + 5.34e4T + 1.35e9T^{2} \)
71 \( 1 + 2.16e4T + 1.80e9T^{2} \)
73 \( 1 + 8.28e4T + 2.07e9T^{2} \)
79 \( 1 - 2.39e4T + 3.07e9T^{2} \)
83 \( 1 + 8.11e4T + 3.93e9T^{2} \)
89 \( 1 - 1.00e5T + 5.58e9T^{2} \)
97 \( 1 - 3.61e4T + 8.58e9T^{2} \)
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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

−9.068034817837854252413760497725, −8.166070466118250460827083245139, −7.55048989378911556948159511763, −6.56739369619575275151812773313, −5.80607700379345855233860528665, −4.71862239829514296638613682037, −4.00075624164073641631254321216, −2.99895821356065109551796385191, −1.60305724514617324391202612185, −0.38656200442501889003190434393, 0.38656200442501889003190434393, 1.60305724514617324391202612185, 2.99895821356065109551796385191, 4.00075624164073641631254321216, 4.71862239829514296638613682037, 5.80607700379345855233860528665, 6.56739369619575275151812773313, 7.55048989378911556948159511763, 8.166070466118250460827083245139, 9.068034817837854252413760497725

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