Properties

Label 2-1104-1.1-c5-0-43
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 − 48.0·5-s + 163.·7-s + 81·9-s − 261.·11-s + 860.·13-s − 432.·15-s + 676.·17-s − 1.93e3·19-s + 1.47e3·21-s + 529·23-s − 819.·25-s + 729·27-s + 4.96e3·29-s + 2.87e3·31-s − 2.35e3·33-s − 7.84e3·35-s + 6.98e3·37-s + 7.74e3·39-s + 1.86e3·41-s − 9.17e3·43-s − 3.88e3·45-s − 6.60e3·47-s + 9.87e3·49-s + 6.08e3·51-s + 2.69e4·53-s + 1.25e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.858·5-s + 1.26·7-s + 0.333·9-s − 0.651·11-s + 1.41·13-s − 0.495·15-s + 0.567·17-s − 1.23·19-s + 0.727·21-s + 0.208·23-s − 0.262·25-s + 0.192·27-s + 1.09·29-s + 0.536·31-s − 0.376·33-s − 1.08·35-s + 0.839·37-s + 0.814·39-s + 0.173·41-s − 0.756·43-s − 0.286·45-s − 0.435·47-s + 0.587·49-s + 0.327·51-s + 1.31·53-s + 0.559·55-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

\(L(3)\) \(\approx\) \(3.035112598\)
\(L(\frac12)\) \(\approx\) \(3.035112598\)
\(L(\frac{7}{2})\) 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 \( 1 + 48.0T + 3.12e3T^{2} \)
7 \( 1 - 163.T + 1.68e4T^{2} \)
11 \( 1 + 261.T + 1.61e5T^{2} \)
13 \( 1 - 860.T + 3.71e5T^{2} \)
17 \( 1 - 676.T + 1.41e6T^{2} \)
19 \( 1 + 1.93e3T + 2.47e6T^{2} \)
29 \( 1 - 4.96e3T + 2.05e7T^{2} \)
31 \( 1 - 2.87e3T + 2.86e7T^{2} \)
37 \( 1 - 6.98e3T + 6.93e7T^{2} \)
41 \( 1 - 1.86e3T + 1.15e8T^{2} \)
43 \( 1 + 9.17e3T + 1.47e8T^{2} \)
47 \( 1 + 6.60e3T + 2.29e8T^{2} \)
53 \( 1 - 2.69e4T + 4.18e8T^{2} \)
59 \( 1 - 2.84e4T + 7.14e8T^{2} \)
61 \( 1 + 1.81e3T + 8.44e8T^{2} \)
67 \( 1 + 1.78e4T + 1.35e9T^{2} \)
71 \( 1 - 2.66e4T + 1.80e9T^{2} \)
73 \( 1 + 4.10e4T + 2.07e9T^{2} \)
79 \( 1 + 3.95e4T + 3.07e9T^{2} \)
83 \( 1 + 6.32e4T + 3.93e9T^{2} \)
89 \( 1 - 1.38e5T + 5.58e9T^{2} \)
97 \( 1 + 1.73e5T + 8.58e9T^{2} \)
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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

−8.704790716350905290608217037836, −8.273019785413405162779121630036, −7.82616613028403538075929442281, −6.76502941080269644819667583020, −5.67471897837723015468255471821, −4.57711144206152279443382363739, −3.96577434949852086323554114395, −2.89514414262353476333810748537, −1.76173241252603531014659675146, −0.75007501852478617904846051216, 0.75007501852478617904846051216, 1.76173241252603531014659675146, 2.89514414262353476333810748537, 3.96577434949852086323554114395, 4.57711144206152279443382363739, 5.67471897837723015468255471821, 6.76502941080269644819667583020, 7.82616613028403538075929442281, 8.273019785413405162779121630036, 8.704790716350905290608217037836

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