Properties

Label 2-91-1.1-c7-0-18
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 22.0·2-s + 71.8·3-s + 358.·4-s + 25.2·5-s − 1.58e3·6-s + 343·7-s − 5.09e3·8-s + 2.97e3·9-s − 556.·10-s + 6.38e3·11-s + 2.58e4·12-s + 2.19e3·13-s − 7.56e3·14-s + 1.81e3·15-s + 6.65e4·16-s + 1.63e4·17-s − 6.57e4·18-s − 2.84e4·19-s + 9.05e3·20-s + 2.46e4·21-s − 1.40e5·22-s + 5.93e4·23-s − 3.66e5·24-s − 7.74e4·25-s − 4.84e4·26-s + 5.69e4·27-s + 1.23e5·28-s + ⋯
L(s)  = 1  − 1.95·2-s + 1.53·3-s + 2.80·4-s + 0.0902·5-s − 2.99·6-s + 0.377·7-s − 3.51·8-s + 1.36·9-s − 0.176·10-s + 1.44·11-s + 4.31·12-s + 0.277·13-s − 0.737·14-s + 0.138·15-s + 4.05·16-s + 0.804·17-s − 2.65·18-s − 0.950·19-s + 0.253·20-s + 0.580·21-s − 2.82·22-s + 1.01·23-s − 5.40·24-s − 0.991·25-s − 0.540·26-s + 0.556·27-s + 1.05·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.727150278\)
\(L(\frac12)\) \(\approx\) \(1.727150278\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 343T \)
13 \( 1 - 2.19e3T \)
good2 \( 1 + 22.0T + 128T^{2} \)
3 \( 1 - 71.8T + 2.18e3T^{2} \)
5 \( 1 - 25.2T + 7.81e4T^{2} \)
11 \( 1 - 6.38e3T + 1.94e7T^{2} \)
17 \( 1 - 1.63e4T + 4.10e8T^{2} \)
19 \( 1 + 2.84e4T + 8.93e8T^{2} \)
23 \( 1 - 5.93e4T + 3.40e9T^{2} \)
29 \( 1 - 2.53e5T + 1.72e10T^{2} \)
31 \( 1 + 1.95e5T + 2.75e10T^{2} \)
37 \( 1 + 1.89e5T + 9.49e10T^{2} \)
41 \( 1 - 1.72e5T + 1.94e11T^{2} \)
43 \( 1 - 1.01e5T + 2.71e11T^{2} \)
47 \( 1 - 2.86e5T + 5.06e11T^{2} \)
53 \( 1 + 1.57e6T + 1.17e12T^{2} \)
59 \( 1 + 5.65e4T + 2.48e12T^{2} \)
61 \( 1 - 2.02e6T + 3.14e12T^{2} \)
67 \( 1 - 1.14e6T + 6.06e12T^{2} \)
71 \( 1 + 1.98e6T + 9.09e12T^{2} \)
73 \( 1 - 5.52e6T + 1.10e13T^{2} \)
79 \( 1 - 1.22e6T + 1.92e13T^{2} \)
83 \( 1 + 6.67e5T + 2.71e13T^{2} \)
89 \( 1 - 1.22e7T + 4.42e13T^{2} \)
97 \( 1 + 8.75e6T + 8.07e13T^{2} \)
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

−12.37702709180429032382625783858, −11.18400915268944621182927559439, −9.994257781872070617623119184361, −9.095829946854018365729938230555, −8.519462621655362921034991259699, −7.57334574423153215694320968129, −6.46351693603888451398455693694, −3.49287209667505932660659997569, −2.14066581243329772853750142319, −1.12709111371301620697117808331, 1.12709111371301620697117808331, 2.14066581243329772853750142319, 3.49287209667505932660659997569, 6.46351693603888451398455693694, 7.57334574423153215694320968129, 8.519462621655362921034991259699, 9.095829946854018365729938230555, 9.994257781872070617623119184361, 11.18400915268944621182927559439, 12.37702709180429032382625783858

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