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Properties

Label 2-112-28.27-c9-0-20
Degree $2$
Conductor $112$
Sign $0.811 - 0.584i$
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  + 207.·3-s − 631. i·5-s + (5.15e3 − 3.71e3i)7-s + 2.34e4·9-s + 8.48e4i·11-s + 7.82e4i·13-s − 1.31e5i·15-s + 6.16e5i·17-s − 4.79e5·19-s + (1.07e6 − 7.71e5i)21-s + 5.99e5i·23-s + 1.55e6·25-s + 7.79e5·27-s + 3.67e6·29-s + 4.75e6·31-s + ⋯
L(s)  = 1  + 1.48·3-s − 0.451i·5-s + (0.811 − 0.584i)7-s + 1.19·9-s + 1.74i·11-s + 0.759i·13-s − 0.668i·15-s + 1.79i·17-s − 0.844·19-s + (1.20 − 0.865i)21-s + 0.446i·23-s + 0.795·25-s + 0.282·27-s + 0.963·29-s + 0.924·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.811 - 0.584i$
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: \((2,\ 112,\ (\ :9/2),\ 0.811 - 0.584i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.012529045\)
\(L(\frac12)\) \(\approx\) \(4.012529045\)
\(L(\frac{11}{2})\) 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 \( 1 + (-5.15e3 + 3.71e3i)T \)
good3 \( 1 - 207.T + 1.96e4T^{2} \)
5 \( 1 + 631. iT - 1.95e6T^{2} \)
11 \( 1 - 8.48e4iT - 2.35e9T^{2} \)
13 \( 1 - 7.82e4iT - 1.06e10T^{2} \)
17 \( 1 - 6.16e5iT - 1.18e11T^{2} \)
19 \( 1 + 4.79e5T + 3.22e11T^{2} \)
23 \( 1 - 5.99e5iT - 1.80e12T^{2} \)
29 \( 1 - 3.67e6T + 1.45e13T^{2} \)
31 \( 1 - 4.75e6T + 2.64e13T^{2} \)
37 \( 1 - 1.17e7T + 1.29e14T^{2} \)
41 \( 1 + 3.88e5iT - 3.27e14T^{2} \)
43 \( 1 + 3.80e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.49e7T + 1.11e15T^{2} \)
53 \( 1 + 1.09e8T + 3.29e15T^{2} \)
59 \( 1 - 2.94e7T + 8.66e15T^{2} \)
61 \( 1 + 1.24e8iT - 1.16e16T^{2} \)
67 \( 1 - 1.52e8iT - 2.72e16T^{2} \)
71 \( 1 - 7.56e7iT - 4.58e16T^{2} \)
73 \( 1 + 2.09e8iT - 5.88e16T^{2} \)
79 \( 1 - 4.17e8iT - 1.19e17T^{2} \)
83 \( 1 - 6.01e8T + 1.86e17T^{2} \)
89 \( 1 - 4.52e8iT - 3.50e17T^{2} \)
97 \( 1 + 7.61e8iT - 7.60e17T^{2} \)
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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

−12.23869547831314287543128230342, −10.65297443799069804398654515915, −9.655482005771080141936528804249, −8.580885234949874452321037231965, −7.893321120753615590004455145269, −6.75295855248702847619393716910, −4.64620113653014219364934808005, −3.96346801078378561751659913669, −2.21897033975595802903637499371, −1.46870395604250175064650086190, 0.870364078575986041019675190817, 2.65355493915312194984036665277, 3.02605800889126059462067304688, 4.73509604510712212175562519909, 6.25019276609546130790543356676, 7.81240448543944637919945443448, 8.438100974916722877379620184896, 9.288119963794495610536631249798, 10.69007389153408041322802836809, 11.63319632116263067305018607828

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