↑

Properties

Label 1-112-112.27-r0-0-0
Degree $1$
Conductor $112$
Sign $-0.382 - 0.923i$
Analytic cond. $0.520125$
Root an. cond. $0.520125$
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  − i·3-s − i·5-s − 9-s − i·11-s + i·13-s − 15-s − 17-s − i·19-s + 23-s − 25-s + i·27-s − i·29-s + 31-s − 33-s + i·37-s + ⋯
L(s)  = 1  − i·3-s − i·5-s − 9-s − i·11-s + i·13-s − 15-s − 17-s − i·19-s + 23-s − 25-s + i·27-s − i·29-s + 31-s − 33-s + i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(0.520125\)
Root analytic conductor: \(0.520125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 112,\ (0:\ ),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5190104816 - 0.7767540777i\)
\(L(\frac12)\) \(\approx\) \(0.5190104816 - 0.7767540777i\)
\(L(1)\) \(\approx\) \(0.8245509788 - 0.5193117243i\)
\(L(1)\) \(\approx\) \(0.8245509788 - 0.5193117243i\)

Euler product

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

−29.72558423155565883818737348742, −28.58474140979756436103894575327, −27.50313887018022612434013485386, −26.76664990166874405741830016210, −25.81219431338083694260366864747, −24.93680669808358249152344519466, −22.9928545408680475656701106032, −22.70952661139529751412062824293, −21.56663056469619869141208186971, −20.51162864233845094970984008578, −19.547049900248133264553362700858, −18.09824903961264985724214285945, −17.279889273093895409049890626009, −15.80929098584573761751079245558, −15.0544412628717394331283854054, −14.21912234372911825947542137116, −12.638028308436391440051830364420, −11.14440194703323902540720237885, −10.39643145950399933576607395080, −9.38126204601217236206889106590, −7.88235690801071974420220583016, −6.492402029192535482367767543771, −5.09857276679580071930394334453, −3.74982197900302381674846027535, −2.52614859787615639196149570702, 0.95440583083808087788258509901, 2.52432013634000416907028744949, 4.4519511828607911217972207146, 5.870145040985107174055441987472, 7.02865518583676216037116968366, 8.45343516246558317363233138980, 9.12639614454421620966515541204, 11.159719522788952829478102150137, 12.00915276340071234452751618091, 13.29996727930137831268798548768, 13.77084627165018342045410597304, 15.48547491955593630335081798038, 16.76148568216737025438627203606, 17.49037855556254891523836285025, 18.882488516071708181966557704053, 19.57430292080512069177088345570, 20.728161141329723245642550526, 21.84776279823434634069388578753, 23.26316655023189557062140061974, 24.24961548457097367846675527583, 24.60103785243527176922300546741, 25.924699456397585631989049519770, 27.02553478297200288518874312825, 28.57541780434474264938900745983, 28.85193244516978836270454980920

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