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Properties

Label 2-112-28.3-c3-0-11
Degree $2$
Conductor $112$
Sign $-0.999 + 0.0114i$
Analytic cond. $6.60821$
Root an. cond. $2.57064$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.64 − 8.04i)3-s + (−17.1 + 9.92i)5-s + (−18.3 − 2.13i)7-s + (−29.6 − 51.4i)9-s + (−10.2 − 5.91i)11-s − 19.9i·13-s + 184. i·15-s + (1.57 + 0.908i)17-s + (−3.66 − 6.34i)19-s + (−102. + 138. i)21-s + (92.2 − 53.2i)23-s + (134. − 233. i)25-s − 300.·27-s − 191.·29-s + (62.5 − 108. i)31-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)3-s + (−1.53 + 0.887i)5-s + (−0.993 − 0.115i)7-s + (−1.09 − 1.90i)9-s + (−0.280 − 0.162i)11-s − 0.426i·13-s + 3.17i·15-s + (0.0224 + 0.0129i)17-s + (−0.0442 − 0.0765i)19-s + (−1.06 + 1.43i)21-s + (0.835 − 0.482i)23-s + (1.07 − 1.86i)25-s − 2.14·27-s − 1.22·29-s + (0.362 − 0.627i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0114i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.999 + 0.0114i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ -0.999 + 0.0114i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00417908 - 0.732051i\)
\(L(\frac12)\) \(\approx\) \(0.00417908 - 0.732051i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.3 + 2.13i)T \)
good3 \( 1 + (-4.64 + 8.04i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (17.1 - 9.92i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (10.2 + 5.91i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 19.9iT - 2.19e3T^{2} \)
17 \( 1 + (-1.57 - 0.908i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (3.66 + 6.34i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-92.2 + 53.2i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 191.T + 2.43e4T^{2} \)
31 \( 1 + (-62.5 + 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (158. + 274. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 321. iT - 6.89e4T^{2} \)
43 \( 1 + 74.3iT - 7.95e4T^{2} \)
47 \( 1 + (-77.2 - 133. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-159. + 276. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (30.5 - 52.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (267. - 154. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (514. + 297. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 48.6iT - 3.57e5T^{2} \)
73 \( 1 + (667. + 385. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-831. + 480. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.06e3T + 5.71e5T^{2} \)
89 \( 1 + (567. - 327. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 704. iT - 9.12e5T^{2} \)
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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

−12.72355729378533496802702366843, −11.86856607455127842771520321045, −10.72668909073057083399677689462, −9.029894599750074064559706980452, −7.85948314764237528927808579933, −7.28365641080867673293279573933, −6.34990238834157539277149446430, −3.61618886731582318603448738050, −2.73241138990969254474995474914, −0.34627046476442821770211882259, 3.21776443116134806282382032703, 4.06937633944397133528100345573, 5.13206548786430281008932886880, 7.40760233601107401263343984553, 8.627934330060544663863904877811, 9.196329361142904846321781153898, 10.32701186032910541041892801667, 11.48675762059825336507450311330, 12.61229575943822387251150536482, 13.75616130175757433348401968034

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