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Properties

Label 2-1104-12.11-c1-0-14
Degree $2$
Conductor $1104$
Sign $0.962 + 0.270i$
Analytic cond. $8.81548$
Root an. cond. $2.96908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.66 − 0.468i)3-s − 0.762i·5-s + 0.762i·7-s + (2.56 + 1.56i)9-s − 1.78·11-s + 0.799·13-s + (−0.357 + 1.27i)15-s − 4.92i·17-s + 7.11i·19-s + (0.357 − 1.27i)21-s + 23-s + 4.41·25-s + (−3.53 − 3.80i)27-s + 9.21i·29-s − 7.02i·31-s + ⋯
L(s)  = 1  + (−0.962 − 0.270i)3-s − 0.340i·5-s + 0.288i·7-s + (0.853 + 0.521i)9-s − 0.538·11-s + 0.221·13-s + (−0.0922 + 0.328i)15-s − 1.19i·17-s + 1.63i·19-s + (0.0779 − 0.277i)21-s + 0.208·23-s + 0.883·25-s + (−0.680 − 0.732i)27-s + 1.71i·29-s − 1.26i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.962 + 0.270i$
Analytic conductor: \(8.81548\)
Root analytic conductor: \(2.96908\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1104} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1104,\ (\ :1/2),\ 0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112780469\)
\(L(\frac12)\) \(\approx\) \(1.112780469\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 0.468i)T \)
23 \( 1 - T \)
good5 \( 1 + 0.762iT - 5T^{2} \)
7 \( 1 - 0.762iT - 7T^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
13 \( 1 - 0.799T + 13T^{2} \)
17 \( 1 + 4.92iT - 17T^{2} \)
19 \( 1 - 7.11iT - 19T^{2} \)
29 \( 1 - 9.21iT - 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 - 3.78T + 37T^{2} \)
41 \( 1 + 5.00iT - 41T^{2} \)
43 \( 1 - 2.95iT - 43T^{2} \)
47 \( 1 - 4.58T + 47T^{2} \)
53 \( 1 + 7.11iT - 53T^{2} \)
59 \( 1 - 5.82T + 59T^{2} \)
61 \( 1 - 5.57T + 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 - 9.78T + 73T^{2} \)
79 \( 1 + 5.58iT - 79T^{2} \)
83 \( 1 - 7.38T + 83T^{2} \)
89 \( 1 - 4.47iT - 89T^{2} \)
97 \( 1 - 5.38T + 97T^{2} \)
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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

−9.923592799689193613203318181940, −9.051652525389621699045843976505, −8.034791753641033537970978863134, −7.28572110040403944149011240433, −6.37267139261746674795317015978, −5.44590956740159932884172006661, −4.93682425185476662970262730827, −3.69027204370765228226535984499, −2.23244874665302969444627516958, −0.848763617561153517973909552204, 0.860081189414698304408183417684, 2.56093560685562495621637352999, 3.87462323503511660961200789928, 4.73065180476101966381487123432, 5.63382040025012463833449087628, 6.53868701128344329178179354242, 7.14841869866582361902399057953, 8.226922446503847869653161330830, 9.191848674140099932864055186611, 10.13397013297789676608633540567

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