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Properties

Label 2-112-28.19-c5-0-11
Degree $2$
Conductor $112$
Sign $0.543 + 0.839i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.30 + 10.9i)3-s + (−63.5 − 36.7i)5-s + (−43.9 + 121. i)7-s + (42.0 − 72.8i)9-s + (192. − 111. i)11-s − 442. i·13-s − 925. i·15-s + (12.4 − 7.20i)17-s + (1.28e3 − 2.23e3i)19-s + (−1.60e3 + 289. i)21-s + (−340. − 196. i)23-s + (1.13e3 + 1.96e3i)25-s + 4.12e3·27-s + 4.13e3·29-s + (−1.70e3 − 2.95e3i)31-s + ⋯
L(s)  = 1  + (0.404 + 0.700i)3-s + (−1.13 − 0.656i)5-s + (−0.338 + 0.940i)7-s + (0.173 − 0.299i)9-s + (0.479 − 0.276i)11-s − 0.726i·13-s − 1.06i·15-s + (0.0104 − 0.00604i)17-s + (0.819 − 1.41i)19-s + (−0.795 + 0.143i)21-s + (−0.134 − 0.0774i)23-s + (0.362 + 0.627i)25-s + 1.08·27-s + 0.912·29-s + (−0.318 − 0.551i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.543 + 0.839i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.543 + 0.839i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.388197686\)
\(L(\frac12)\) \(\approx\) \(1.388197686\)
\(L(\frac{7}{2})\) 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 \( 1 + (43.9 - 121. i)T \)
good3 \( 1 + (-6.30 - 10.9i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (63.5 + 36.7i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-192. + 111. i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 442. iT - 3.71e5T^{2} \)
17 \( 1 + (-12.4 + 7.20i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.28e3 + 2.23e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (340. + 196. i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.13e3T + 2.05e7T^{2} \)
31 \( 1 + (1.70e3 + 2.95e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (5.39e3 - 9.34e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.92e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.98e4iT - 1.47e8T^{2} \)
47 \( 1 + (-6.07e3 + 1.05e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-818. - 1.41e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.18e4 - 3.78e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (3.97e3 + 2.29e3i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (4.34e4 - 2.51e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 2.05e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.03e4 + 5.94e3i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (6.44e4 + 3.72e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 6.21e3T + 3.93e9T^{2} \)
89 \( 1 + (6.19e4 + 3.57e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 2.84e4iT - 8.58e9T^{2} \)
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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

−12.25160377126648825116323706167, −11.75252520866570492840764119191, −10.27596250506874198845112484137, −9.025252725173774066578606793707, −8.556665589204642533551513604086, −7.06183496181024508581226163588, −5.38674815423808055232126265349, −4.12661762952826781597356658607, −3.01298044439683178575572330424, −0.54802635472212425529721232151, 1.38357255675297329700266207576, 3.25357278221638599499712891919, 4.35827577055617690257550157869, 6.56609214260419046515445402964, 7.40354621925290466585512599125, 8.085103028316365560253747228543, 9.728258520423816416206482414133, 10.84944557918416230090825114128, 11.87412045236651505396676005137, 12.82068923509520607510015539458

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