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Properties

Label 2-112-28.19-c5-0-1
Degree $2$
Conductor $112$
Sign $0.0702 - 0.997i$
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  + (−4.47 − 7.75i)3-s + (−62.1 − 35.8i)5-s + (−115. − 58.3i)7-s + (81.4 − 141. i)9-s + (132. − 76.7i)11-s + 891. i·13-s + 642. i·15-s + (−1.31e3 + 760. i)17-s + (884. − 1.53e3i)19-s + (65.4 + 1.15e3i)21-s + (3.32e3 + 1.91e3i)23-s + (1.01e3 + 1.75e3i)25-s − 3.63e3·27-s − 2.65e3·29-s + (3.08e3 + 5.35e3i)31-s + ⋯
L(s)  = 1  + (−0.287 − 0.497i)3-s + (−1.11 − 0.641i)5-s + (−0.892 − 0.450i)7-s + (0.335 − 0.580i)9-s + (0.331 − 0.191i)11-s + 1.46i·13-s + 0.737i·15-s + (−1.10 + 0.638i)17-s + (0.561 − 0.973i)19-s + (0.0323 + 0.573i)21-s + (1.31 + 0.756i)23-s + (0.323 + 0.560i)25-s − 0.959·27-s − 0.587·29-s + (0.577 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0702 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.0702 - 0.997i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.0702 - 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3292035559\)
\(L(\frac12)\) \(\approx\) \(0.3292035559\)
\(L(\frac{7}{2})\) 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 \( 1 + (115. + 58.3i)T \)
good3 \( 1 + (4.47 + 7.75i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (62.1 + 35.8i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-132. + 76.7i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 891. iT - 3.71e5T^{2} \)
17 \( 1 + (1.31e3 - 760. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-884. + 1.53e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-3.32e3 - 1.91e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.65e3T + 2.05e7T^{2} \)
31 \( 1 + (-3.08e3 - 5.35e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (4.26e3 - 7.38e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.87e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.92e4iT - 1.47e8T^{2} \)
47 \( 1 + (3.54e3 - 6.13e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.25e4 + 2.17e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (645. + 1.11e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-2.07e4 - 1.19e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.42e4 - 8.23e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.58e4iT - 1.80e9T^{2} \)
73 \( 1 + (-9.04e3 + 5.22e3i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.48e4 - 8.59e3i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 5.39e4T + 3.93e9T^{2} \)
89 \( 1 + (2.65e4 + 1.53e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 7.02e4iT - 8.58e9T^{2} \)
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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

−12.88170474440155604530738449738, −11.87984089957627048936357036820, −11.21968967827393783725420879415, −9.523569446564557516184561414013, −8.694915484840031013770039627795, −7.11363334835585695220237285398, −6.58830497006685845632009187241, −4.61160079804396549096534666849, −3.54512242471722906413265272337, −1.17915566440750619698287608913, 0.14707489161659741299116164877, 2.82886514834802103834448141242, 4.03037024697039290221431897686, 5.47059544553283125980009129424, 6.93149640588907720588836707733, 7.920255288483304201301530519770, 9.358022931064297023976105828744, 10.48392741015203484605546678240, 11.23488092160319173675204040581, 12.38018032039999701855022846769

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