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Properties

Label 2-112-28.27-c9-0-10
Degree $2$
Conductor $112$
Sign $0.994 - 0.103i$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 188.·3-s − 2.17e3i·5-s + (6.31e3 − 657. i)7-s + 1.59e4·9-s + 2.61e4i·11-s + 1.07e5i·13-s + 4.11e5i·15-s − 3.32e5i·17-s − 3.52e5·19-s + (−1.19e6 + 1.24e5i)21-s + 6.29e5i·23-s − 2.78e6·25-s + 6.96e5·27-s − 5.19e6·29-s + 3.99e6·31-s + ⋯
L(s)  = 1  − 1.34·3-s − 1.55i·5-s + (0.994 − 0.103i)7-s + 0.812·9-s + 0.538i·11-s + 1.04i·13-s + 2.09i·15-s − 0.965i·17-s − 0.620·19-s + (−1.33 + 0.139i)21-s + 0.469i·23-s − 1.42·25-s + 0.252·27-s − 1.36·29-s + 0.776·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.994 - 0.103i$
Analytic conductor: \(57.6840\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ 0.994 - 0.103i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.095531375\)
\(L(\frac12)\) \(\approx\) \(1.095531375\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-6.31e3 + 657. i)T \)
good3 \( 1 + 188.T + 1.96e4T^{2} \)
5 \( 1 + 2.17e3iT - 1.95e6T^{2} \)
11 \( 1 - 2.61e4iT - 2.35e9T^{2} \)
13 \( 1 - 1.07e5iT - 1.06e10T^{2} \)
17 \( 1 + 3.32e5iT - 1.18e11T^{2} \)
19 \( 1 + 3.52e5T + 3.22e11T^{2} \)
23 \( 1 - 6.29e5iT - 1.80e12T^{2} \)
29 \( 1 + 5.19e6T + 1.45e13T^{2} \)
31 \( 1 - 3.99e6T + 2.64e13T^{2} \)
37 \( 1 + 3.51e6T + 1.29e14T^{2} \)
41 \( 1 - 2.94e7iT - 3.27e14T^{2} \)
43 \( 1 - 3.88e7iT - 5.02e14T^{2} \)
47 \( 1 - 2.28e7T + 1.11e15T^{2} \)
53 \( 1 + 2.09e7T + 3.29e15T^{2} \)
59 \( 1 - 6.51e7T + 8.66e15T^{2} \)
61 \( 1 - 1.31e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.25e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.42e8iT - 4.58e16T^{2} \)
73 \( 1 + 1.86e8iT - 5.88e16T^{2} \)
79 \( 1 + 1.84e8iT - 1.19e17T^{2} \)
83 \( 1 - 4.96e8T + 1.86e17T^{2} \)
89 \( 1 - 5.36e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.13e9iT - 7.60e17T^{2} \)
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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

−11.74368449411585884964562133943, −11.23246959526704404001274964962, −9.731724458587708509597633463613, −8.710708098882415910620985247442, −7.44491359792628474152631108789, −6.02110027355506465559144632014, −4.85606939210061442398728538907, −4.51846349662912553547258805073, −1.74140820078005375174256228920, −0.792862166572124194039907469275, 0.48394301570325391221366461654, 2.19294598439212079991398745407, 3.73980878855377340302602527753, 5.35662080425442478491660407642, 6.13285603942067578453906575060, 7.20199249914913380264948682037, 8.408737107297014821609991618367, 10.47600755247708682964621396677, 10.70330797153874038668342216295, 11.52879866722070561322577488972

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