↑

Properties

Label 2-112-28.3-c5-0-16
Degree $2$
Conductor $112$
Sign $-0.828 + 0.559i$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.47 − 7.75i)3-s + (−62.1 + 35.8i)5-s + (115. − 58.3i)7-s + (81.4 + 141. i)9-s + (−132. − 76.7i)11-s − 891. i·13-s + 642. i·15-s + (−1.31e3 − 760. i)17-s + (−884. − 1.53e3i)19-s + (65.4 − 1.15e3i)21-s + (−3.32e3 + 1.91e3i)23-s + (1.01e3 − 1.75e3i)25-s + 3.63e3·27-s − 2.65e3·29-s + (−3.08e3 + 5.35e3i)31-s + ⋯
L(s)  = 1  + (0.287 − 0.497i)3-s + (−1.11 + 0.641i)5-s + (0.892 − 0.450i)7-s + (0.335 + 0.580i)9-s + (−0.331 − 0.191i)11-s − 1.46i·13-s + 0.737i·15-s + (−1.10 − 0.638i)17-s + (−0.561 − 0.973i)19-s + (0.0323 − 0.573i)21-s + (−1.31 + 0.756i)23-s + (0.323 − 0.560i)25-s + 0.959·27-s − 0.587·29-s + (−0.577 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.828 + 0.559i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ -0.828 + 0.559i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7828768404\)
\(L(\frac12)\) \(\approx\) \(0.7828768404\)
\(L(\frac{7}{2})\) 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 \( 1 + (-115. + 58.3i)T \)
good3 \( 1 + (-4.47 + 7.75i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (62.1 - 35.8i)T + (1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (132. + 76.7i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 891. iT - 3.71e5T^{2} \)
17 \( 1 + (1.31e3 + 760. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (884. + 1.53e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (3.32e3 - 1.91e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 2.65e3T + 2.05e7T^{2} \)
31 \( 1 + (3.08e3 - 5.35e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (4.26e3 + 7.38e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.87e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.92e4iT - 1.47e8T^{2} \)
47 \( 1 + (-3.54e3 - 6.13e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.25e4 - 2.17e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-645. + 1.11e3i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.07e4 + 1.19e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.42e4 - 8.23e3i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.58e4iT - 1.80e9T^{2} \)
73 \( 1 + (-9.04e3 - 5.22e3i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.48e4 - 8.59e3i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 5.39e4T + 3.93e9T^{2} \)
89 \( 1 + (2.65e4 - 1.53e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 7.02e4iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.26812152573833026658770969452, −10.99804445573901952079646127182, −10.61991639032839237595359142721, −8.638927028786713336531306305423, −7.64107670578993004082940687792, −7.17342497702380461205127137994, −5.19038335287268560680039966741, −3.79662767061424779176436573923, −2.24005111618191167093167790549, −0.27313286842062289305911388499, 1.83777893742736201285858677325, 4.04785721954096725543765304490, 4.52973986108838533135353148036, 6.40489565467711459980144592725, 8.001573483549452804622101020561, 8.646858061136733646145499267600, 9.786398165840946575613218022803, 11.23532043882337289681681894993, 11.96336748991817436715320668592, 12.90397941407866760507051799937

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