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Properties

Label 2-112-7.4-c7-0-16
Degree $2$
Conductor $112$
Sign $0.749 - 0.661i$
Analytic cond. $34.9871$
Root an. cond. $5.91499$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−38.4 + 66.5i)3-s + (10.6 + 18.3i)5-s + (642. + 641. i)7-s + (−1.86e3 − 3.22e3i)9-s + (1.63e3 − 2.82e3i)11-s + 1.02e4·13-s − 1.63e3·15-s + (1.82e4 − 3.16e4i)17-s + (−1.62e4 − 2.80e4i)19-s + (−6.73e4 + 1.81e4i)21-s + (8.37e3 + 1.44e4i)23-s + (3.88e4 − 6.72e4i)25-s + 1.18e5·27-s + 4.39e4·29-s + (6.69e4 − 1.15e5i)31-s + ⋯
L(s)  = 1  + (−0.821 + 1.42i)3-s + (0.0380 + 0.0658i)5-s + (0.707 + 0.706i)7-s + (−0.851 − 1.47i)9-s + (0.369 − 0.640i)11-s + 1.29·13-s − 0.124·15-s + (0.901 − 1.56i)17-s + (−0.541 − 0.938i)19-s + (−1.58 + 0.426i)21-s + (0.143 + 0.248i)23-s + (0.497 − 0.861i)25-s + 1.15·27-s + 0.334·29-s + (0.403 − 0.698i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(34.9871\)
Root analytic conductor: \(5.91499\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :7/2),\ 0.749 - 0.661i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.749882719\)
\(L(\frac12)\) \(\approx\) \(1.749882719\)
\(L(\frac{9}{2})\) 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 \( 1 + (-642. - 641. i)T \)
good3 \( 1 + (38.4 - 66.5i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-10.6 - 18.3i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-1.63e3 + 2.82e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 - 1.02e4T + 6.27e7T^{2} \)
17 \( 1 + (-1.82e4 + 3.16e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (1.62e4 + 2.80e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-8.37e3 - 1.44e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 4.39e4T + 1.72e10T^{2} \)
31 \( 1 + (-6.69e4 + 1.15e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-2.89e5 - 5.02e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 5.32e5T + 1.94e11T^{2} \)
43 \( 1 - 3.65e5T + 2.71e11T^{2} \)
47 \( 1 + (1.03e5 + 1.79e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-5.44e5 + 9.42e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-8.08e4 + 1.39e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-4.22e5 - 7.31e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-6.13e5 + 1.06e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 1.10e6T + 9.09e12T^{2} \)
73 \( 1 + (8.19e5 - 1.41e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (2.49e6 + 4.32e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 3.38e6T + 2.71e13T^{2} \)
89 \( 1 + (3.26e6 + 5.66e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 1.35e6T + 8.07e13T^{2} \)
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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

−11.71910030278796229593438810923, −11.44249197003673510670693729482, −10.36904755838601086065407309242, −9.281019543889157324244091557515, −8.372014162274931954028927240635, −6.39615836360413559016135767672, −5.36706936313943651802250696248, −4.45216819025686938335354811119, −3.03332456989281096420639097977, −0.76568639539233903143506167619, 1.07285812465913162608659216288, 1.66792146921959299221710171537, 3.97496804500164084641777848236, 5.62010687581688194441933080990, 6.54634827366561084924101940239, 7.61144447498558929901340757068, 8.485600288181441565317525714810, 10.42526976507782221858042501806, 11.17307671120388037049521930875, 12.31337598722485002999978428126

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