Properties

Label 2-1104-1.1-c1-0-1
Degree $2$
Conductor $1104$
Sign $1$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4.42·5-s − 0.622·7-s + 9-s − 5.80·11-s + 2·13-s + 4.42·15-s + 1.37·17-s − 6.42·19-s + 0.622·21-s + 23-s + 14.6·25-s − 27-s − 0.755·29-s + 1.24·31-s + 5.80·33-s + 2.75·35-s + 9.05·37-s − 2·39-s − 9.61·41-s + 5.18·43-s − 4.42·45-s + 5.24·47-s − 6.61·49-s − 1.37·51-s + 12.0·53-s + 25.7·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.98·5-s − 0.235·7-s + 0.333·9-s − 1.75·11-s + 0.554·13-s + 1.14·15-s + 0.334·17-s − 1.47·19-s + 0.135·21-s + 0.208·23-s + 2.92·25-s − 0.192·27-s − 0.140·29-s + 0.223·31-s + 1.01·33-s + 0.465·35-s + 1.48·37-s − 0.320·39-s − 1.50·41-s + 0.790·43-s − 0.660·45-s + 0.764·47-s − 0.944·49-s − 0.192·51-s + 1.65·53-s + 3.46·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $1$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5126783215\)
\(L(\frac12)\) \(\approx\) \(0.5126783215\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + 4.42T + 5T^{2} \)
7 \( 1 + 0.622T + 7T^{2} \)
11 \( 1 + 5.80T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.37T + 17T^{2} \)
19 \( 1 + 6.42T + 19T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 - 9.05T + 37T^{2} \)
41 \( 1 + 9.61T + 41T^{2} \)
43 \( 1 - 5.18T + 43T^{2} \)
47 \( 1 - 5.24T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 - 2.94T + 61T^{2} \)
67 \( 1 + 5.18T + 67T^{2} \)
71 \( 1 + 6.10T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8.62T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 - 6.23T + 89T^{2} \)
97 \( 1 + 2.85T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19928258656506714078037117707, −8.718651993633261235742664101418, −8.120686891753311086327396512929, −7.46955464318567561692288637246, −6.62623392310406154252545215878, −5.47955888416803786877964722493, −4.54397226023140078323647354917, −3.79086309884720451163885921534, −2.68544767092925493409204352717, −0.53551138840216696430331899125, 0.53551138840216696430331899125, 2.68544767092925493409204352717, 3.79086309884720451163885921534, 4.54397226023140078323647354917, 5.47955888416803786877964722493, 6.62623392310406154252545215878, 7.46955464318567561692288637246, 8.120686891753311086327396512929, 8.718651993633261235742664101418, 10.19928258656506714078037117707

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