↑

Properties

Label 1-112-112.5-r1-0-0
Degree $1$
Conductor $112$
Sign $0.868 - 0.496i$
Analytic cond. $12.0360$
Root an. cond. $12.0360$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s − i·13-s + 15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.866 − 0.5i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s − i·13-s + 15-s + (0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.866 − 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(12.0360\)
Root analytic conductor: \(12.0360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 112,\ (1:\ ),\ 0.868 - 0.496i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9099248612 - 0.2417914452i\)
\(L(\frac12)\) \(\approx\) \(0.9099248612 - 0.2417914452i\)
\(L(1)\) \(\approx\) \(0.7317987211 - 0.07650563618i\)
\(L(1)\) \(\approx\) \(0.7317987211 - 0.07650563618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.89338311961100512564160161501, −28.24313696130447858641565219335, −27.32833157906085437580061253160, −26.51904377136176015131657429801, −25.06964951820124497554296764087, −23.78800232031377029932868281706, −23.17658545880913945346118846269, −22.2202992181505183671637332469, −20.89978440237786869789562867395, −20.180209402529715431794297399, −18.709738618231591831745703350603, −17.659588336012879631573094573510, −16.52763655103130756071328196627, −15.71230705742770030246914349760, −14.82181599371302953821222694488, −12.81475664747785088387406751184, −12.22187793536348269317483450843, −10.89221215768217568180799491234, −10.03509985519932443703177649544, −8.45412015977234867726002751458, −7.276937527746327638082527656934, −5.62683869819182595358956888348, −4.69969071184107238585882748143, −3.354040133380914889154632821070, −0.85746469006092104261567041411, 0.68196650161367119556087592690, 2.73302369535162686617911218198, 4.44804315400907501299744634832, 5.77920505935299304265475897983, 7.125544445372640967746579371708, 7.88164985200290861064568040886, 9.71795245864979331976592585763, 11.2816646248368443380964032315, 11.55739360555560567076022700473, 13.003238311403890620502002524610, 14.11980878099498317695882193816, 15.699258187647013786223172635097, 16.35035787493786292297513775449, 17.72790256889625173335728492879, 18.73294870077625517501172428027, 19.33221441795333329134385579827, 20.93787139471294476236617953720, 22.1048117348635508736759503134, 23.09635578039961449039536422248, 23.75745795895877424435987771481, 24.706231029767630368336427566224, 26.233926358519581770175136976441, 27.065545270865520648782974272945, 28.17157942686630816760651673159, 29.06652825942199535641320381366

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