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Properties

Label 2-112-28.19-c5-0-9
Degree $2$
Conductor $112$
Sign $0.723 - 0.690i$
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  + (4.74 + 8.21i)3-s + (45.4 + 26.2i)5-s + (128. + 20.4i)7-s + (76.5 − 132. i)9-s + (21.1 − 12.2i)11-s − 58.8i·13-s + 497. i·15-s + (334. − 192. i)17-s + (−292. + 506. i)19-s + (439. + 1.14e3i)21-s + (2.81e3 + 1.62e3i)23-s + (−186. − 323. i)25-s + 3.75e3·27-s − 6.05e3·29-s + (4.75e3 + 8.23e3i)31-s + ⋯
L(s)  = 1  + (0.304 + 0.526i)3-s + (0.812 + 0.469i)5-s + (0.987 + 0.157i)7-s + (0.315 − 0.545i)9-s + (0.0527 − 0.0304i)11-s − 0.0965i·13-s + 0.570i·15-s + (0.280 − 0.161i)17-s + (−0.185 + 0.321i)19-s + (0.217 + 0.568i)21-s + (1.10 + 0.639i)23-s + (−0.0597 − 0.103i)25-s + 0.991·27-s − 1.33·29-s + (0.888 + 1.53i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.723 - 0.690i$
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: \((2,\ 112,\ (\ :5/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.745391658\)
\(L(\frac12)\) \(\approx\) \(2.745391658\)
\(L(\frac{7}{2})\) 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 \( 1 + (-128. - 20.4i)T \)
good3 \( 1 + (-4.74 - 8.21i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-45.4 - 26.2i)T + (1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-21.1 + 12.2i)T + (8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 58.8iT - 3.71e5T^{2} \)
17 \( 1 + (-334. + 192. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (292. - 506. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-2.81e3 - 1.62e3i)T + (3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 6.05e3T + 2.05e7T^{2} \)
31 \( 1 + (-4.75e3 - 8.23e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-2.89e3 + 5.01e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 8.85e3iT - 1.15e8T^{2} \)
43 \( 1 - 7.39e3iT - 1.47e8T^{2} \)
47 \( 1 + (-1.31e3 + 2.27e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (6.07e3 + 1.05e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-7.26e3 - 1.25e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.81e4 + 1.04e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (3.65e4 - 2.11e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.51e4iT - 1.80e9T^{2} \)
73 \( 1 + (8.33e3 - 4.81e3i)T + (1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (7.18e4 + 4.14e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 8.10e4T + 3.93e9T^{2} \)
89 \( 1 + (2.68e4 + 1.54e4i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.31e5iT - 8.58e9T^{2} \)
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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

−12.89217246951146715289905946053, −11.63833530994517809263784941309, −10.57118728774711303737609237118, −9.653274149006696132208636472556, −8.675007316591846794970435698886, −7.28811855543547883973492126491, −5.93384939116223267361483534507, −4.64665892626577426099288783816, −3.11108871194385541589865102085, −1.50088556292640578200012472829, 1.22134632964292705922141264988, 2.30525061136902183829327318914, 4.48358690396583729859079502069, 5.62299251178427059338424128687, 7.15353173138522228640150268601, 8.156284635841822318574168323874, 9.221111090687076928153216393724, 10.46099593370881376944904587088, 11.52042084567668484935870311521, 12.87115022420041640676965050544

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