Properties

Label 2-91-1.1-c1-0-1
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 1.41·3-s + 1.58·5-s − 2.00·6-s + 7-s + 2.82·8-s − 0.999·9-s − 2.24·10-s + 4.24·11-s − 13-s − 1.41·14-s + 2.24·15-s − 4.00·16-s + 1.41·17-s + 1.41·18-s − 7.24·19-s + 1.41·21-s − 6·22-s − 5.82·23-s + 4·24-s − 2.48·25-s + 1.41·26-s − 5.65·27-s + 0.171·29-s − 3.17·30-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.816·3-s + 0.709·5-s − 0.816·6-s + 0.377·7-s + 0.999·8-s − 0.333·9-s − 0.709·10-s + 1.27·11-s − 0.277·13-s − 0.377·14-s + 0.579·15-s − 1.00·16-s + 0.342·17-s + 0.333·18-s − 1.66·19-s + 0.308·21-s − 1.27·22-s − 1.21·23-s + 0.816·24-s − 0.497·25-s + 0.277·26-s − 1.08·27-s + 0.0318·29-s − 0.579·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8035041043\)
\(L(\frac12)\) \(\approx\) \(0.8035041043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
13 \( 1 + T \)
good2 \( 1 + 1.41T + 2T^{2} \)
3 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 - 1.58T + 5T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 + 7.24T + 19T^{2} \)
23 \( 1 + 5.82T + 23T^{2} \)
29 \( 1 - 0.171T + 29T^{2} \)
31 \( 1 - 3.24T + 31T^{2} \)
37 \( 1 - 2.24T + 37T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 + 0.171T + 53T^{2} \)
59 \( 1 - 0.343T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 - 1.58T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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

−14.19259351354014001633935862578, −13.32961004431274155578181993754, −11.81034674227379139892165090587, −10.43811267915924677562981769248, −9.453975216742547325282993143478, −8.710892224506815951873187739116, −7.78956849606875471561960327198, −6.15614991745813169223168872051, −4.20245225198679567841491642802, −1.98957828776268363334345914285, 1.98957828776268363334345914285, 4.20245225198679567841491642802, 6.15614991745813169223168872051, 7.78956849606875471561960327198, 8.710892224506815951873187739116, 9.453975216742547325282993143478, 10.43811267915924677562981769248, 11.81034674227379139892165090587, 13.32961004431274155578181993754, 14.19259351354014001633935862578

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