↑

Properties

Label 2-112-28.27-c9-0-15
Degree $2$
Conductor $112$
Sign $0.567 - 0.823i$
Analytic cond. $57.6840$
Root an. cond. $7.59499$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 270.·3-s + 701. i·5-s + (2.72e3 + 5.73e3i)7-s + 5.33e4·9-s + 4.57e3i·11-s + 1.11e5i·13-s − 1.89e5i·15-s − 4.93e5i·17-s − 3.94e5·19-s + (−7.37e5 − 1.55e6i)21-s − 1.10e6i·23-s + 1.46e6·25-s − 9.10e6·27-s + 5.63e6·29-s − 2.51e6·31-s + ⋯
L(s)  = 1  − 1.92·3-s + 0.502i·5-s + (0.429 + 0.903i)7-s + 2.71·9-s + 0.0941i·11-s + 1.07i·13-s − 0.967i·15-s − 1.43i·17-s − 0.694·19-s + (−0.827 − 1.73i)21-s − 0.826i·23-s + 0.747·25-s − 3.29·27-s + 1.47·29-s − 0.489·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(57.6840\)
Root analytic conductor: \(7.59499\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :9/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.023246779\)
\(L(\frac12)\) \(\approx\) \(1.023246779\)
\(L(\frac{11}{2})\) 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 \( 1 + (-2.72e3 - 5.73e3i)T \)
good3 \( 1 + 270.T + 1.96e4T^{2} \)
5 \( 1 - 701. iT - 1.95e6T^{2} \)
11 \( 1 - 4.57e3iT - 2.35e9T^{2} \)
13 \( 1 - 1.11e5iT - 1.06e10T^{2} \)
17 \( 1 + 4.93e5iT - 1.18e11T^{2} \)
19 \( 1 + 3.94e5T + 3.22e11T^{2} \)
23 \( 1 + 1.10e6iT - 1.80e12T^{2} \)
29 \( 1 - 5.63e6T + 1.45e13T^{2} \)
31 \( 1 + 2.51e6T + 2.64e13T^{2} \)
37 \( 1 + 1.25e6T + 1.29e14T^{2} \)
41 \( 1 + 2.67e7iT - 3.27e14T^{2} \)
43 \( 1 - 7.37e6iT - 5.02e14T^{2} \)
47 \( 1 - 2.22e7T + 1.11e15T^{2} \)
53 \( 1 - 7.63e7T + 3.29e15T^{2} \)
59 \( 1 - 1.04e8T + 8.66e15T^{2} \)
61 \( 1 - 1.23e8iT - 1.16e16T^{2} \)
67 \( 1 + 3.27e8iT - 2.72e16T^{2} \)
71 \( 1 + 6.83e7iT - 4.58e16T^{2} \)
73 \( 1 + 1.39e8iT - 5.88e16T^{2} \)
79 \( 1 - 5.83e8iT - 1.19e17T^{2} \)
83 \( 1 - 2.35e7T + 1.86e17T^{2} \)
89 \( 1 - 1.91e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.26e9iT - 7.60e17T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.90172128217640529632678829934, −11.12635727244133364409353348300, −10.27362535094001771661500854206, −8.963777811829723213308959465825, −7.12050369913463987411401386608, −6.42528401979496827315557298682, −5.27916897990654682515899183767, −4.43362210090486737733237547816, −2.24340519253973966270172987597, −0.72071266887543478959516310576, 0.58797932888003958801552776947, 1.36490519211784650676411849586, 4.01668188489606388149400454080, 4.99345188789939226375404989051, 5.94696703140526217564398444101, 7.02020051142638929795898791502, 8.264842589937981344406080329023, 10.17668050160631725933235643056, 10.62636301987717135289955381765, 11.59415593001898746825209814903

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