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Properties

Label 2-112-28.3-c5-0-8
Degree $2$
Conductor $112$
Sign $0.785 + 0.619i$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.5 + 20.0i)3-s + (−75.7 + 43.7i)5-s + (−90.8 − 92.4i)7-s + (−145. − 252. i)9-s + (−350. − 202. i)11-s + 640. i·13-s − 2.02e3i·15-s + (1.28e3 + 742. i)17-s + (699. + 1.21e3i)19-s + (2.90e3 − 750. i)21-s + (−1.03e3 + 600. i)23-s + (2.26e3 − 3.92e3i)25-s + 1.12e3·27-s − 1.96e3·29-s + (406. − 704. i)31-s + ⋯
L(s)  = 1  + (−0.741 + 1.28i)3-s + (−1.35 + 0.782i)5-s + (−0.700 − 0.713i)7-s + (−0.600 − 1.03i)9-s + (−0.874 − 0.504i)11-s + 1.05i·13-s − 2.32i·15-s + (1.07 + 0.623i)17-s + (0.444 + 0.770i)19-s + (1.43 − 0.371i)21-s + (−0.409 + 0.236i)23-s + (0.725 − 1.25i)25-s + 0.297·27-s − 0.434·29-s + (0.0760 − 0.131i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.785 + 0.619i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.2016162261\)
\(L(\frac12)\) \(\approx\) \(0.2016162261\)
\(L(\frac{7}{2})\) 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 \( 1 + (90.8 + 92.4i)T \)
good3 \( 1 + (11.5 - 20.0i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (75.7 - 43.7i)T + (1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (350. + 202. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 640. iT - 3.71e5T^{2} \)
17 \( 1 + (-1.28e3 - 742. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-699. - 1.21e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.03e3 - 600. i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 1.96e3T + 2.05e7T^{2} \)
31 \( 1 + (-406. + 704. i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-6.95e3 - 1.20e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.70e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.92e4iT - 1.47e8T^{2} \)
47 \( 1 + (1.13e4 + 1.96e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-8.00e3 + 1.38e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.50e4 + 2.60e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (3.78e4 - 2.18e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (9.18e3 + 5.30e3i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 5.74e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.63e4 - 3.25e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.59e4 - 9.23e3i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.70e4T + 3.93e9T^{2} \)
89 \( 1 + (4.08e4 - 2.35e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.28e4iT - 8.58e9T^{2} \)
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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

−12.15879613902417977798713007164, −11.34307214851366117965067771134, −10.48053609182547450650844657688, −9.844925668004445876350671740732, −8.125293412916316679993834911841, −6.97712833406893361412298935542, −5.57147018669103497312005448584, −4.02835137391441875615354021515, −3.46949669321813288041613896626, −0.11919999723168575218500401989, 0.853407836479243738119918291101, 2.92944265267246549738417291501, 4.92734394354863407394919374841, 6.01518600563565645725459548134, 7.53654390618534100024499572964, 7.931308934774107859819256578895, 9.505168346615848636288698914297, 11.13567017501481449613452765767, 12.08542030184561838039364777233, 12.61459432445867930142802765773

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