Properties

Label 2-1104-1.1-c5-0-13
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 − 59.5·5-s − 96.8·7-s + 81·9-s + 731.·11-s − 631.·13-s + 536.·15-s + 75.7·17-s − 2.35·19-s + 871.·21-s − 529·23-s + 426.·25-s − 729·27-s − 2.42e3·29-s + 349.·31-s − 6.57e3·33-s + 5.77e3·35-s + 8.31e3·37-s + 5.68e3·39-s + 4.19e3·41-s − 9.37e3·43-s − 4.82e3·45-s + 2.43e3·47-s − 7.43e3·49-s − 682.·51-s − 3.49e4·53-s − 4.35e4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.06·5-s − 0.746·7-s + 0.333·9-s + 1.82·11-s − 1.03·13-s + 0.615·15-s + 0.0636·17-s − 0.00149·19-s + 0.431·21-s − 0.208·23-s + 0.136·25-s − 0.192·27-s − 0.535·29-s + 0.0653·31-s − 1.05·33-s + 0.796·35-s + 0.998·37-s + 0.598·39-s + 0.390·41-s − 0.773·43-s − 0.355·45-s + 0.160·47-s − 0.442·49-s − 0.0367·51-s − 1.71·53-s − 1.94·55-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

\(L(3)\) \(\approx\) \(0.7174861383\)
\(L(\frac12)\) \(\approx\) \(0.7174861383\)
\(L(\frac{7}{2})\) 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 \( 1 + 59.5T + 3.12e3T^{2} \)
7 \( 1 + 96.8T + 1.68e4T^{2} \)
11 \( 1 - 731.T + 1.61e5T^{2} \)
13 \( 1 + 631.T + 3.71e5T^{2} \)
17 \( 1 - 75.7T + 1.41e6T^{2} \)
19 \( 1 + 2.35T + 2.47e6T^{2} \)
29 \( 1 + 2.42e3T + 2.05e7T^{2} \)
31 \( 1 - 349.T + 2.86e7T^{2} \)
37 \( 1 - 8.31e3T + 6.93e7T^{2} \)
41 \( 1 - 4.19e3T + 1.15e8T^{2} \)
43 \( 1 + 9.37e3T + 1.47e8T^{2} \)
47 \( 1 - 2.43e3T + 2.29e8T^{2} \)
53 \( 1 + 3.49e4T + 4.18e8T^{2} \)
59 \( 1 + 1.50e4T + 7.14e8T^{2} \)
61 \( 1 - 987.T + 8.44e8T^{2} \)
67 \( 1 + 4.43e4T + 1.35e9T^{2} \)
71 \( 1 - 3.39e4T + 1.80e9T^{2} \)
73 \( 1 + 4.24e4T + 2.07e9T^{2} \)
79 \( 1 + 5.11e4T + 3.07e9T^{2} \)
83 \( 1 - 8.84e4T + 3.93e9T^{2} \)
89 \( 1 + 3.40e4T + 5.58e9T^{2} \)
97 \( 1 + 1.56e5T + 8.58e9T^{2} \)
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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

−9.393907622603216119329303776671, −8.203456149290317264159514069101, −7.33013335444510881547836701361, −6.65732323933415876610543518610, −5.89914971619276617766446569131, −4.61711917622901085771339976858, −3.98433908224491395392204978632, −3.07980916118966027516888421171, −1.56329030299192855646615698978, −0.38207684130995514944966549318, 0.38207684130995514944966549318, 1.56329030299192855646615698978, 3.07980916118966027516888421171, 3.98433908224491395392204978632, 4.61711917622901085771339976858, 5.89914971619276617766446569131, 6.65732323933415876610543518610, 7.33013335444510881547836701361, 8.203456149290317264159514069101, 9.393907622603216119329303776671

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