Properties

Label 2-112-16.11-c4-0-21
Degree $2$
Conductor $112$
Sign $0.273 - 0.961i$
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  + (−2.74 + 2.90i)2-s + (1.91 + 1.91i)3-s + (−0.925 − 15.9i)4-s + (33.8 + 33.8i)5-s + (−10.8 + 0.312i)6-s + 18.5·7-s + (49.0 + 41.1i)8-s − 73.6i·9-s + (−191. + 5.54i)10-s + (100. − 100. i)11-s + (28.7 − 32.2i)12-s + (28.6 − 28.6i)13-s + (−50.8 + 53.8i)14-s + 129. i·15-s + (−254. + 29.5i)16-s + 549.·17-s + ⋯
L(s)  = 1  + (−0.686 + 0.727i)2-s + (0.212 + 0.212i)3-s + (−0.0578 − 0.998i)4-s + (1.35 + 1.35i)5-s + (−0.300 + 0.00868i)6-s + 0.377·7-s + (0.765 + 0.643i)8-s − 0.909i·9-s + (−1.91 + 0.0554i)10-s + (0.832 − 0.832i)11-s + (0.199 − 0.224i)12-s + (0.169 − 0.169i)13-s + (−0.259 + 0.274i)14-s + 0.574i·15-s + (−0.993 + 0.115i)16-s + 1.90·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.273 - 0.961i$
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: \((2,\ 112,\ (\ :2),\ 0.273 - 0.961i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.46920 + 1.10981i\)
\(L(\frac12)\) \(\approx\) \(1.46920 + 1.10981i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.74 - 2.90i)T \)
7 \( 1 - 18.5T \)
good3 \( 1 + (-1.91 - 1.91i)T + 81iT^{2} \)
5 \( 1 + (-33.8 - 33.8i)T + 625iT^{2} \)
11 \( 1 + (-100. + 100. i)T - 1.46e4iT^{2} \)
13 \( 1 + (-28.6 + 28.6i)T - 2.85e4iT^{2} \)
17 \( 1 - 549.T + 8.35e4T^{2} \)
19 \( 1 + (-58.0 - 58.0i)T + 1.30e5iT^{2} \)
23 \( 1 + 874.T + 2.79e5T^{2} \)
29 \( 1 + (307. - 307. i)T - 7.07e5iT^{2} \)
31 \( 1 - 763. iT - 9.23e5T^{2} \)
37 \( 1 + (292. + 292. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.39e3iT - 2.82e6T^{2} \)
43 \( 1 + (-757. + 757. i)T - 3.41e6iT^{2} \)
47 \( 1 - 494. iT - 4.87e6T^{2} \)
53 \( 1 + (-203. - 203. i)T + 7.89e6iT^{2} \)
59 \( 1 + (-3.35e3 + 3.35e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (4.49e3 - 4.49e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (5.20e3 + 5.20e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 6.52e3T + 2.54e7T^{2} \)
73 \( 1 + 1.86e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.14e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.75e3 + 1.75e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 1.37e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.95e3T + 8.85e7T^{2} \)
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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

−13.78330363912696545897513753006, −11.82662790866724444583169514916, −10.53175567625671346141063719484, −9.888164142323606629674614078453, −8.966388607609066448742360628284, −7.55588055707009592982106951246, −6.28276863140373056785190467630, −5.75447523370592006928242086848, −3.33157817066288970077506696821, −1.44992567916742748866939885313, 1.28134979824314144970034395873, 2.09446440201507923680703325995, 4.34951024573256786977785979402, 5.69251449257364469393829164225, 7.60529090580165182963577998234, 8.563405998066653398975656530841, 9.624327435882077675762373238329, 10.18402611418123990598890529082, 11.84642916506267012389066796060, 12.54630717811729678919414980014

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