Properties

Label 2-91-1.1-c7-0-0
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  − 17.3·2-s − 89.1·3-s + 173.·4-s − 9.67·5-s + 1.54e3·6-s − 343·7-s − 784.·8-s + 5.75e3·9-s + 167.·10-s − 4.81e3·11-s − 1.54e4·12-s − 2.19e3·13-s + 5.95e3·14-s + 862.·15-s − 8.55e3·16-s − 1.32e4·17-s − 9.99e4·18-s − 2.16e4·19-s − 1.67e3·20-s + 3.05e4·21-s + 8.34e4·22-s − 8.74e4·23-s + 6.99e4·24-s − 7.80e4·25-s + 3.81e4·26-s − 3.18e5·27-s − 5.94e4·28-s + ⋯
L(s)  = 1  − 1.53·2-s − 1.90·3-s + 1.35·4-s − 0.0346·5-s + 2.92·6-s − 0.377·7-s − 0.541·8-s + 2.63·9-s + 0.0530·10-s − 1.08·11-s − 2.57·12-s − 0.277·13-s + 0.579·14-s + 0.0659·15-s − 0.521·16-s − 0.652·17-s − 4.03·18-s − 0.725·19-s − 0.0468·20-s + 0.720·21-s + 1.67·22-s − 1.49·23-s + 1.03·24-s − 0.998·25-s + 0.425·26-s − 3.11·27-s − 0.511·28-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: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.02788899752\)
\(L(\frac12)\) \(\approx\) \(0.02788899752\)
\(L(\frac{9}{2})\) 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 \( 1 + 17.3T + 128T^{2} \)
3 \( 1 + 89.1T + 2.18e3T^{2} \)
5 \( 1 + 9.67T + 7.81e4T^{2} \)
11 \( 1 + 4.81e3T + 1.94e7T^{2} \)
17 \( 1 + 1.32e4T + 4.10e8T^{2} \)
19 \( 1 + 2.16e4T + 8.93e8T^{2} \)
23 \( 1 + 8.74e4T + 3.40e9T^{2} \)
29 \( 1 - 1.95e5T + 1.72e10T^{2} \)
31 \( 1 + 1.66e5T + 2.75e10T^{2} \)
37 \( 1 - 5.25e4T + 9.49e10T^{2} \)
41 \( 1 + 4.40e5T + 1.94e11T^{2} \)
43 \( 1 + 6.97e5T + 2.71e11T^{2} \)
47 \( 1 + 4.17e4T + 5.06e11T^{2} \)
53 \( 1 + 1.88e6T + 1.17e12T^{2} \)
59 \( 1 + 3.45e3T + 2.48e12T^{2} \)
61 \( 1 + 1.64e5T + 3.14e12T^{2} \)
67 \( 1 + 9.80e4T + 6.06e12T^{2} \)
71 \( 1 - 1.68e6T + 9.09e12T^{2} \)
73 \( 1 + 6.15e6T + 1.10e13T^{2} \)
79 \( 1 + 1.71e5T + 1.92e13T^{2} \)
83 \( 1 + 7.99e6T + 2.71e13T^{2} \)
89 \( 1 - 5.25e6T + 4.42e13T^{2} \)
97 \( 1 + 5.31e6T + 8.07e13T^{2} \)
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

−12.20953019503828590342288361296, −11.25584704673034164108744999224, −10.36410195170605598235872847705, −9.840513413058647138061469128417, −8.157462554479898743415817590893, −6.99085379333460602103789231705, −6.02207675142202267993555627569, −4.61872304528981445914895013324, −1.81125503792889124325968887100, −0.14246684585066868369796002863, 0.14246684585066868369796002863, 1.81125503792889124325968887100, 4.61872304528981445914895013324, 6.02207675142202267993555627569, 6.99085379333460602103789231705, 8.157462554479898743415817590893, 9.840513413058647138061469128417, 10.36410195170605598235872847705, 11.25584704673034164108744999224, 12.20953019503828590342288361296

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