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Properties

Label 2-112-112.93-c1-0-3
Degree $2$
Conductor $112$
Sign $0.989 - 0.141i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.743 − 1.20i)2-s + (−0.222 + 0.831i)3-s + (−0.893 + 1.78i)4-s + (0.543 + 2.02i)5-s + (1.16 − 0.350i)6-s + (2.63 − 0.233i)7-s + (2.81 − 0.256i)8-s + (1.95 + 1.12i)9-s + (2.03 − 2.16i)10-s + (−3.85 − 1.03i)11-s + (−1.28 − 1.14i)12-s + (−0.990 − 0.990i)13-s + (−2.24 − 2.99i)14-s − 1.80·15-s + (−2.40 − 3.19i)16-s + (3.07 + 5.33i)17-s + ⋯
L(s)  = 1  + (−0.526 − 0.850i)2-s + (−0.128 + 0.480i)3-s + (−0.446 + 0.894i)4-s + (0.242 + 0.906i)5-s + (0.476 − 0.143i)6-s + (0.996 − 0.0881i)7-s + (0.995 − 0.0907i)8-s + (0.652 + 0.376i)9-s + (0.643 − 0.683i)10-s + (−1.16 − 0.311i)11-s + (−0.372 − 0.329i)12-s + (−0.274 − 0.274i)13-s + (−0.598 − 0.800i)14-s − 0.466·15-s + (−0.601 − 0.799i)16-s + (0.746 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.989 - 0.141i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.989 - 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837750 + 0.0595863i\)
\(L(\frac12)\) \(\approx\) \(0.837750 + 0.0595863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.743 + 1.20i)T \)
7 \( 1 + (-2.63 + 0.233i)T \)
good3 \( 1 + (0.222 - 0.831i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.543 - 2.02i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (3.85 + 1.03i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.990 + 0.990i)T + 13iT^{2} \)
17 \( 1 + (-3.07 - 5.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.79 + 1.01i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.91 + 3.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.83 + 3.83i)T + 29iT^{2} \)
31 \( 1 + (2.05 + 3.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0198 + 0.0740i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + (-0.713 + 0.713i)T - 43iT^{2} \)
47 \( 1 + (-1.95 + 3.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.06 + 1.89i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.17 - 0.851i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-8.84 + 2.37i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.401 + 1.49i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.86iT - 71T^{2} \)
73 \( 1 + (8.95 - 5.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.33 - 5.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.2 - 10.2i)T + 83iT^{2} \)
89 \( 1 + (-1.16 - 0.671i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−13.54942873452661323550711869312, −12.44655958029728100703659371317, −11.16274406726403740789950568247, −10.48892203926023409044427279300, −9.915654113416621893301748921821, −8.210900062398689615365557321112, −7.47341672679587777661841313864, −5.38510082631356935964211934961, −3.90976923546893748417346927231, −2.23057484592445011265337659882, 1.43914882210682040621484860034, 4.79558874199732427916093646313, 5.56455746824283318763984028402, 7.31293417210131988256602885433, 7.88372806547435953107796008333, 9.231294682258237382987719629518, 10.07310555329371200360925159561, 11.59937840621340040858907711492, 12.71995519235005879814941951692, 13.71912038701128917805506760176

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