Properties

Label 2-91-1.1-c7-0-16
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.80·2-s − 76.5·3-s − 113.·4-s − 339.·5-s + 290.·6-s + 343·7-s + 917.·8-s + 3.67e3·9-s + 1.29e3·10-s + 3.39e3·11-s + 8.69e3·12-s − 2.19e3·13-s − 1.30e3·14-s + 2.59e4·15-s + 1.10e4·16-s + 8.09e3·17-s − 1.39e4·18-s − 4.96e4·19-s + 3.85e4·20-s − 2.62e4·21-s − 1.29e4·22-s + 2.06e4·23-s − 7.02e4·24-s + 3.71e4·25-s + 8.34e3·26-s − 1.13e5·27-s − 3.89e4·28-s + ⋯
L(s)  = 1  − 0.335·2-s − 1.63·3-s − 0.887·4-s − 1.21·5-s + 0.549·6-s + 0.377·7-s + 0.633·8-s + 1.67·9-s + 0.407·10-s + 0.770·11-s + 1.45·12-s − 0.277·13-s − 0.126·14-s + 1.98·15-s + 0.674·16-s + 0.399·17-s − 0.563·18-s − 1.66·19-s + 1.07·20-s − 0.618·21-s − 0.258·22-s + 0.353·23-s − 1.03·24-s + 0.475·25-s + 0.0931·26-s − 1.11·27-s − 0.335·28-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: \(1\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 3.80T + 128T^{2} \)
3 \( 1 + 76.5T + 2.18e3T^{2} \)
5 \( 1 + 339.T + 7.81e4T^{2} \)
11 \( 1 - 3.39e3T + 1.94e7T^{2} \)
17 \( 1 - 8.09e3T + 4.10e8T^{2} \)
19 \( 1 + 4.96e4T + 8.93e8T^{2} \)
23 \( 1 - 2.06e4T + 3.40e9T^{2} \)
29 \( 1 - 1.38e5T + 1.72e10T^{2} \)
31 \( 1 - 9.25e4T + 2.75e10T^{2} \)
37 \( 1 - 4.04e4T + 9.49e10T^{2} \)
41 \( 1 - 5.24e4T + 1.94e11T^{2} \)
43 \( 1 - 6.96e5T + 2.71e11T^{2} \)
47 \( 1 - 7.73e4T + 5.06e11T^{2} \)
53 \( 1 + 6.82e5T + 1.17e12T^{2} \)
59 \( 1 + 1.73e6T + 2.48e12T^{2} \)
61 \( 1 - 2.70e6T + 3.14e12T^{2} \)
67 \( 1 + 4.71e6T + 6.06e12T^{2} \)
71 \( 1 - 2.79e6T + 9.09e12T^{2} \)
73 \( 1 + 3.16e6T + 1.10e13T^{2} \)
79 \( 1 - 2.13e6T + 1.92e13T^{2} \)
83 \( 1 - 5.08e6T + 2.71e13T^{2} \)
89 \( 1 + 1.26e7T + 4.42e13T^{2} \)
97 \( 1 - 1.03e7T + 8.07e13T^{2} \)
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

−12.03717797773810890765819843767, −11.12201132046029967455700527574, −10.22658292179547027813625033623, −8.742902181345339337182100826966, −7.58312240079118369304208126311, −6.26296463379932050325315901245, −4.80037475847453748240447208417, −4.10703250810770656976087154394, −0.995886737109739991821878512394, 0, 0.995886737109739991821878512394, 4.10703250810770656976087154394, 4.80037475847453748240447208417, 6.26296463379932050325315901245, 7.58312240079118369304208126311, 8.742902181345339337182100826966, 10.22658292179547027813625033623, 11.12201132046029967455700527574, 12.03717797773810890765819843767

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