Properties

Label 2-112-16.11-c4-0-9
Degree $2$
Conductor $112$
Sign $0.262 - 0.964i$
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  + (−2.91 − 2.73i)2-s + (0.212 + 0.212i)3-s + (1.01 + 15.9i)4-s + (14.9 + 14.9i)5-s + (−0.0380 − 1.20i)6-s + 18.5·7-s + (40.7 − 49.3i)8-s − 80.9i·9-s + (−2.68 − 84.7i)10-s + (−139. + 139. i)11-s + (−3.17 + 3.60i)12-s + (−186. + 186. i)13-s + (−54.0 − 50.6i)14-s + 6.36i·15-s + (−253. + 32.3i)16-s + 167.·17-s + ⋯
L(s)  = 1  + (−0.729 − 0.684i)2-s + (0.0236 + 0.0236i)3-s + (0.0633 + 0.997i)4-s + (0.599 + 0.599i)5-s + (−0.00105 − 0.0333i)6-s + 0.377·7-s + (0.636 − 0.771i)8-s − 0.998i·9-s + (−0.0268 − 0.847i)10-s + (−1.15 + 1.15i)11-s + (−0.0220 + 0.0250i)12-s + (−1.10 + 1.10i)13-s + (−0.275 − 0.258i)14-s + 0.0283i·15-s + (−0.991 + 0.126i)16-s + 0.578·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.262 - 0.964i$
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: \((2,\ 112,\ (\ :2),\ 0.262 - 0.964i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.735629 + 0.562033i\)
\(L(\frac12)\) \(\approx\) \(0.735629 + 0.562033i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.91 + 2.73i)T \)
7 \( 1 - 18.5T \)
good3 \( 1 + (-0.212 - 0.212i)T + 81iT^{2} \)
5 \( 1 + (-14.9 - 14.9i)T + 625iT^{2} \)
11 \( 1 + (139. - 139. i)T - 1.46e4iT^{2} \)
13 \( 1 + (186. - 186. i)T - 2.85e4iT^{2} \)
17 \( 1 - 167.T + 8.35e4T^{2} \)
19 \( 1 + (-146. - 146. i)T + 1.30e5iT^{2} \)
23 \( 1 - 367.T + 2.79e5T^{2} \)
29 \( 1 + (1.04e3 - 1.04e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.41e3iT - 9.23e5T^{2} \)
37 \( 1 + (-173. - 173. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.18e3iT - 2.82e6T^{2} \)
43 \( 1 + (316. - 316. i)T - 3.41e6iT^{2} \)
47 \( 1 - 2.99e3iT - 4.87e6T^{2} \)
53 \( 1 + (1.38e3 + 1.38e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (3.09e3 - 3.09e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-1.84e3 + 1.84e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.02e3 + 2.02e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 8.04e3T + 2.54e7T^{2} \)
73 \( 1 + 5.96e3iT - 2.83e7T^{2} \)
79 \( 1 + 5.16e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.17e3 - 7.17e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 6.31e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.11e4T + 8.85e7T^{2} \)
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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

−12.71699917998632536326970741268, −12.09151927724363062222795490592, −10.83505126233333907391332731768, −9.890412674253198804077376603575, −9.234468027016523232706289273927, −7.65084548392088514879305384464, −6.78997286880567445727046152953, −4.82912322584733151941329485463, −3.05402928027346112636511479935, −1.73800548014640285468160893353, 0.50230005286290772033235234581, 2.35051449543301510823761173492, 5.23873824802176394893002910959, 5.53531872251240897907396266280, 7.57939155413932087084955926303, 8.099560256475313256985641127696, 9.407517243334925686168736832132, 10.38032022379288664570200951283, 11.28598320378555422548428841218, 13.10262697258918179552543432354

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