Properties

Label 2-112-16.11-c4-0-4
Degree $2$
Conductor $112$
Sign $-0.983 - 0.181i$
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  + (−3.77 − 1.33i)2-s + (10.8 + 10.8i)3-s + (12.4 + 10.0i)4-s + (−31.7 − 31.7i)5-s + (−26.4 − 55.3i)6-s + 18.5·7-s + (−33.4 − 54.5i)8-s + 154. i·9-s + (77.2 + 161. i)10-s + (−126. + 126. i)11-s + (25.6 + 244. i)12-s + (−44.9 + 44.9i)13-s + (−69.8 − 24.7i)14-s − 688. i·15-s + (53.2 + 250. i)16-s − 110.·17-s + ⋯
L(s)  = 1  + (−0.942 − 0.333i)2-s + (1.20 + 1.20i)3-s + (0.777 + 0.629i)4-s + (−1.26 − 1.26i)5-s + (−0.734 − 1.53i)6-s + 0.377·7-s + (−0.522 − 0.852i)8-s + 1.90i·9-s + (0.772 + 1.61i)10-s + (−1.04 + 1.04i)11-s + (0.178 + 1.69i)12-s + (−0.266 + 0.266i)13-s + (−0.356 − 0.126i)14-s − 3.05i·15-s + (0.208 + 0.978i)16-s − 0.381·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.983 - 0.181i$
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: \((2,\ 112,\ (\ :2),\ -0.983 - 0.181i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0331339 + 0.361145i\)
\(L(\frac12)\) \(\approx\) \(0.0331339 + 0.361145i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.77 + 1.33i)T \)
7 \( 1 - 18.5T \)
good3 \( 1 + (-10.8 - 10.8i)T + 81iT^{2} \)
5 \( 1 + (31.7 + 31.7i)T + 625iT^{2} \)
11 \( 1 + (126. - 126. i)T - 1.46e4iT^{2} \)
13 \( 1 + (44.9 - 44.9i)T - 2.85e4iT^{2} \)
17 \( 1 + 110.T + 8.35e4T^{2} \)
19 \( 1 + (125. + 125. i)T + 1.30e5iT^{2} \)
23 \( 1 + 915.T + 2.79e5T^{2} \)
29 \( 1 + (459. - 459. i)T - 7.07e5iT^{2} \)
31 \( 1 + 1.61e3iT - 9.23e5T^{2} \)
37 \( 1 + (-964. - 964. i)T + 1.87e6iT^{2} \)
41 \( 1 + 936. iT - 2.82e6T^{2} \)
43 \( 1 + (895. - 895. i)T - 3.41e6iT^{2} \)
47 \( 1 - 1.58e3iT - 4.87e6T^{2} \)
53 \( 1 + (909. + 909. i)T + 7.89e6iT^{2} \)
59 \( 1 + (950. - 950. i)T - 1.21e7iT^{2} \)
61 \( 1 + (658. - 658. i)T - 1.38e7iT^{2} \)
67 \( 1 + (2.97e3 + 2.97e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 7.20e3T + 2.54e7T^{2} \)
73 \( 1 - 2.35e3iT - 2.83e7T^{2} \)
79 \( 1 - 341. iT - 3.89e7T^{2} \)
83 \( 1 + (-6.30e3 - 6.30e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 4.32e3iT - 6.27e7T^{2} \)
97 \( 1 - 3.63e3T + 8.85e7T^{2} \)
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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

−13.21639611286799591887912511871, −12.18878554320541838157127506468, −11.08066115437803910600730526038, −9.886695691725685852069867434792, −9.141091111712405980514725373384, −8.081688502759343398341249514411, −7.75013465313629691455823170744, −4.71665403896290649763462667224, −3.91561111070483755863279886711, −2.22586104092154198286875566646, 0.17201951918845306178318034629, 2.23580852406533321322473369287, 3.32100688644434964532049153326, 6.24447455869894652884514649805, 7.37062379973693752530528959322, 7.950175140457611699075662710407, 8.543179763111761569584097845155, 10.29830348982299024533817942918, 11.25981156813709541408176628497, 12.28445103367842104171991768063

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