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Properties

Label 2-1104-12.11-c1-0-15
Degree $2$
Conductor $1104$
Sign $0.158 - 0.987i$
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  + (1.61 − 0.618i)3-s + 3.68i·5-s + 3.18i·7-s + (2.23 − 2.00i)9-s + 0.732·11-s − 1.46·13-s + (2.27 + 5.96i)15-s − 5.51i·17-s + 3.68i·19-s + (1.96 + 5.15i)21-s + 23-s − 8.60·25-s + (2.38 − 4.61i)27-s + 4.47i·29-s + 5.60i·31-s + ⋯
L(s)  = 1  + (0.934 − 0.356i)3-s + 1.64i·5-s + 1.20i·7-s + (0.745 − 0.666i)9-s + 0.220·11-s − 0.406·13-s + (0.588 + 1.54i)15-s − 1.33i·17-s + 0.846i·19-s + (0.429 + 1.12i)21-s + 0.208·23-s − 1.72·25-s + (0.458 − 0.888i)27-s + 0.830i·29-s + 1.00i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1104\)    =    \(2^{4} \cdot 3 \cdot 23\)
Sign: $0.158 - 0.987i$
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: \((2,\ 1104,\ (\ :1/2),\ 0.158 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.156339974\)
\(L(\frac12)\) \(\approx\) \(2.156339974\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.618i)T \)
23 \( 1 - T \)
good5 \( 1 - 3.68iT - 5T^{2} \)
7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 - 0.732T + 11T^{2} \)
13 \( 1 + 1.46T + 13T^{2} \)
17 \( 1 + 5.51iT - 17T^{2} \)
19 \( 1 - 3.68iT - 19T^{2} \)
29 \( 1 - 4.47iT - 29T^{2} \)
31 \( 1 - 5.60iT - 31T^{2} \)
37 \( 1 - 6.87T + 37T^{2} \)
41 \( 1 - 2.70iT - 41T^{2} \)
43 \( 1 - 8.52iT - 43T^{2} \)
47 \( 1 + 9.60T + 47T^{2} \)
53 \( 1 + 9.04iT - 53T^{2} \)
59 \( 1 + 7.70T + 59T^{2} \)
61 \( 1 + 9.80T + 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 - 2.10T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 + 8.58iT - 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 1.97iT - 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−9.773085501821801663017223569264, −9.360238461356452362254606557316, −8.299424351872975501821608560862, −7.53404955355776173706241726085, −6.77053736516619018430380836644, −6.11518423705839690503188540194, −4.82415358827403125852533483206, −3.27355471571109146941099045117, −2.90479019528004614299120461059, −1.90030030615202056133719056211, 0.874494949181688045860336581277, 2.08720044280991142472037119858, 3.69722091753640602303052733566, 4.33928692667401911780300181557, 4.99289407197534391805911164155, 6.30484803443790057656418038825, 7.59377033974344384583267150601, 8.013100365359242381744712298986, 8.950750614285427012688677610137, 9.462545963121255362725263865502

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