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Properties

Label 2-112-112.83-c3-0-12
Degree $2$
Conductor $112$
Sign $0.927 - 0.372i$
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  + (−2.35 − 1.56i)2-s + (4.79 + 4.79i)3-s + (3.10 + 7.37i)4-s + (−2.04 − 2.04i)5-s + (−3.79 − 18.8i)6-s + (13.3 − 12.8i)7-s + (4.23 − 22.2i)8-s + 19.0i·9-s + (1.61 + 8.02i)10-s + (34.5 + 34.5i)11-s + (−20.4 + 50.2i)12-s + (7.32 − 7.32i)13-s + (−51.5 + 9.25i)14-s − 19.6i·15-s + (−44.7 + 45.7i)16-s + 133. i·17-s + ⋯
L(s)  = 1  + (−0.832 − 0.553i)2-s + (0.923 + 0.923i)3-s + (0.387 + 0.921i)4-s + (−0.183 − 0.183i)5-s + (−0.258 − 1.27i)6-s + (0.722 − 0.691i)7-s + (0.187 − 0.982i)8-s + 0.704i·9-s + (0.0511 + 0.253i)10-s + (0.947 + 0.947i)11-s + (−0.493 + 1.20i)12-s + (0.156 − 0.156i)13-s + (−0.984 + 0.176i)14-s − 0.337i·15-s + (−0.699 + 0.714i)16-s + 1.90i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.927 - 0.372i$
Analytic conductor: \(6.60821\)
Root analytic conductor: \(2.57064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :3/2),\ 0.927 - 0.372i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.54139 + 0.298183i\)
\(L(\frac12)\) \(\approx\) \(1.54139 + 0.298183i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.35 + 1.56i)T \)
7 \( 1 + (-13.3 + 12.8i)T \)
good3 \( 1 + (-4.79 - 4.79i)T + 27iT^{2} \)
5 \( 1 + (2.04 + 2.04i)T + 125iT^{2} \)
11 \( 1 + (-34.5 - 34.5i)T + 1.33e3iT^{2} \)
13 \( 1 + (-7.32 + 7.32i)T - 2.19e3iT^{2} \)
17 \( 1 - 133. iT - 4.91e3T^{2} \)
19 \( 1 + (-13.6 - 13.6i)T + 6.85e3iT^{2} \)
23 \( 1 - 129.T + 1.21e4T^{2} \)
29 \( 1 + (-85.7 - 85.7i)T + 2.43e4iT^{2} \)
31 \( 1 + 239.T + 2.97e4T^{2} \)
37 \( 1 + (50.7 - 50.7i)T - 5.06e4iT^{2} \)
41 \( 1 + 118.T + 6.89e4T^{2} \)
43 \( 1 + (75.4 + 75.4i)T + 7.95e4iT^{2} \)
47 \( 1 + 169.T + 1.03e5T^{2} \)
53 \( 1 + (-482. + 482. i)T - 1.48e5iT^{2} \)
59 \( 1 + (285. - 285. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-313. + 313. i)T - 2.26e5iT^{2} \)
67 \( 1 + (58.2 - 58.2i)T - 3.00e5iT^{2} \)
71 \( 1 + 374.T + 3.57e5T^{2} \)
73 \( 1 + 610.T + 3.89e5T^{2} \)
79 \( 1 + 460. iT - 4.93e5T^{2} \)
83 \( 1 + (875. + 875. i)T + 5.71e5iT^{2} \)
89 \( 1 - 633.T + 7.04e5T^{2} \)
97 \( 1 - 946. iT - 9.12e5T^{2} \)
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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

−13.07540228963580102448681506396, −11.98040109463818052982747304547, −10.70712419734493943802695157509, −10.05407486495291382593470474376, −8.899442568965888806371698230285, −8.246198200771023297114148151581, −6.94835128983409133340841155781, −4.39936684754480236970688843566, −3.56080378712965016692446519024, −1.61975678780513645508736827148, 1.24804694543421308641610485851, 2.78321493033912323608937205703, 5.32139018168570109555980626090, 6.83963212810693717840955757492, 7.62052326874658968599639088659, 8.776904831169447031509435395252, 9.206791855967804786878960951640, 11.11526920157505201266588408614, 11.78578860602843970110096597355, 13.46783860306222299243182592537

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