Properties

Label 2-112-16.11-c4-0-22
Degree $2$
Conductor $112$
Sign $0.744 + 0.667i$
Analytic cond. $11.5774$
Root an. cond. $3.40256$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.55 − 1.83i)2-s + (4.91 + 4.91i)3-s + (9.25 + 13.0i)4-s + (−5.29 − 5.29i)5-s + (−8.43 − 26.4i)6-s − 18.5·7-s + (−8.89 − 63.3i)8-s − 32.6i·9-s + (9.09 + 28.5i)10-s + (65.1 − 65.1i)11-s + (−18.6 + 109. i)12-s + (2.97 − 2.97i)13-s + (65.8 + 34.0i)14-s − 52.0i·15-s + (−84.8 + 241. i)16-s + 212.·17-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)2-s + (0.546 + 0.546i)3-s + (0.578 + 0.815i)4-s + (−0.211 − 0.211i)5-s + (−0.234 − 0.736i)6-s − 0.377·7-s + (−0.139 − 0.990i)8-s − 0.403i·9-s + (0.0909 + 0.285i)10-s + (0.538 − 0.538i)11-s + (−0.129 + 0.761i)12-s + (0.0176 − 0.0176i)13-s + (0.335 + 0.173i)14-s − 0.231i·15-s + (−0.331 + 0.943i)16-s + 0.735·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.744 + 0.667i$
Analytic conductor: \(11.5774\)
Root analytic conductor: \(3.40256\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :2),\ 0.744 + 0.667i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.22927 - 0.469974i\)
\(L(\frac12)\) \(\approx\) \(1.22927 - 0.469974i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.55 + 1.83i)T \)
7 \( 1 + 18.5T \)
good3 \( 1 + (-4.91 - 4.91i)T + 81iT^{2} \)
5 \( 1 + (5.29 + 5.29i)T + 625iT^{2} \)
11 \( 1 + (-65.1 + 65.1i)T - 1.46e4iT^{2} \)
13 \( 1 + (-2.97 + 2.97i)T - 2.85e4iT^{2} \)
17 \( 1 - 212.T + 8.35e4T^{2} \)
19 \( 1 + (-301. - 301. i)T + 1.30e5iT^{2} \)
23 \( 1 - 474.T + 2.79e5T^{2} \)
29 \( 1 + (-355. + 355. i)T - 7.07e5iT^{2} \)
31 \( 1 + 801. iT - 9.23e5T^{2} \)
37 \( 1 + (-151. - 151. i)T + 1.87e6iT^{2} \)
41 \( 1 + 788. iT - 2.82e6T^{2} \)
43 \( 1 + (-853. + 853. i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.38e3iT - 4.87e6T^{2} \)
53 \( 1 + (-1.52e3 - 1.52e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-3.61e3 + 3.61e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.23e3 - 4.23e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.92e3 + 2.92e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 2.71e3T + 2.54e7T^{2} \)
73 \( 1 - 803. iT - 2.83e7T^{2} \)
79 \( 1 + 1.03e4iT - 3.89e7T^{2} \)
83 \( 1 + (-322. - 322. i)T + 4.74e7iT^{2} \)
89 \( 1 - 8.47e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.16e3T + 8.85e7T^{2} \)
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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

−12.43610163418598402436188276977, −11.71836233807722111014873007778, −10.39091662344350129574688528189, −9.523111431744863772642226693077, −8.739299293484470906027083057560, −7.64926693831375322028188417074, −6.19017516262833029051977778740, −3.99097605674541956570074301581, −2.97857706392736186081040183264, −0.870690395764043827897788922374, 1.28932368812285723018622148750, 2.91541974085375059935148022495, 5.21113662991466125957610821760, 6.86554795426136883088935791719, 7.44317929153479511309083314455, 8.652006452728181692975377742757, 9.587286696947491700275756276083, 10.74490250514803319114563473537, 11.86293029637521775807765798557, 13.17388996257833294048104294288

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