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Properties

Label 2-112-7.4-c3-0-4
Degree $2$
Conductor $112$
Sign $0.874 - 0.485i$
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  + (−0.5 + 0.866i)3-s + (−3.5 − 6.06i)5-s + (10 + 15.5i)7-s + (13 + 22.5i)9-s + (17.5 − 30.3i)11-s + 66·13-s + 7·15-s + (−29.5 + 51.0i)17-s + (68.5 + 118. i)19-s + (−18.5 + 0.866i)21-s + (−3.5 − 6.06i)23-s + (38 − 65.8i)25-s − 53·27-s + 106·29-s + (37.5 − 64.9i)31-s + ⋯
L(s)  = 1  + (−0.0962 + 0.166i)3-s + (−0.313 − 0.542i)5-s + (0.539 + 0.841i)7-s + (0.481 + 0.833i)9-s + (0.479 − 0.830i)11-s + 1.40·13-s + 0.120·15-s + (−0.420 + 0.728i)17-s + (0.827 + 1.43i)19-s + (−0.192 + 0.00899i)21-s + (−0.0317 − 0.0549i)23-s + (0.303 − 0.526i)25-s − 0.377·27-s + 0.678·29-s + (0.217 − 0.376i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.60639 + 0.416176i\)
\(L(\frac12)\) \(\approx\) \(1.60639 + 0.416176i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10 - 15.5i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (3.5 + 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-17.5 + 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 66T + 2.19e3T^{2} \)
17 \( 1 + (29.5 - 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-68.5 - 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 106T + 2.43e4T^{2} \)
31 \( 1 + (-37.5 + 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (5.5 + 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 + 260T + 7.95e4T^{2} \)
47 \( 1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-208.5 + 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (8.5 - 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (25.5 + 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-219.5 + 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 784T + 3.57e5T^{2} \)
73 \( 1 + (147.5 - 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (247.5 + 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 - 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−13.24831628589291895815022951191, −12.07160688257089399611132425111, −11.25085703868421261012768432946, −10.14349508903992616982421448462, −8.551759326689261201310840605209, −8.212789062632414080876327895333, −6.28330267556535683809957265939, −5.14072807366062122748080176405, −3.74014406165420185107643424478, −1.54602903899650683507618455726, 1.14992680721516712133275645612, 3.44622426064655006654250058404, 4.72235878389007995424466189618, 6.69326887992141851398920707387, 7.21086197163421905529926249460, 8.749883203989109086502694423882, 9.943992475412367864589102768047, 11.13613888674052675324148817839, 11.80150690184479113187725869763, 13.20216117057822084934598657134

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