Properties

Label 2-91-1.1-c9-0-25
Degree $2$
Conductor $91$
Sign $-1$
Analytic cond. $46.8682$
Root an. cond. $6.84603$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 13.0·2-s − 81.8·3-s − 342.·4-s − 1.04e3·5-s + 1.06e3·6-s + 2.40e3·7-s + 1.11e4·8-s − 1.29e4·9-s + 1.35e4·10-s − 5.26e4·11-s + 2.80e4·12-s + 2.85e4·13-s − 3.12e4·14-s + 8.53e4·15-s + 3.09e4·16-s + 6.16e5·17-s + 1.68e5·18-s + 2.62e5·19-s + 3.57e5·20-s − 1.96e5·21-s + 6.84e5·22-s + 1.18e6·23-s − 9.10e5·24-s − 8.65e5·25-s − 3.71e5·26-s + 2.67e6·27-s − 8.23e5·28-s + ⋯
L(s)  = 1  − 0.574·2-s − 0.583·3-s − 0.669·4-s − 0.746·5-s + 0.335·6-s + 0.377·7-s + 0.959·8-s − 0.659·9-s + 0.428·10-s − 1.08·11-s + 0.390·12-s + 0.277·13-s − 0.217·14-s + 0.435·15-s + 0.118·16-s + 1.79·17-s + 0.379·18-s + 0.461·19-s + 0.499·20-s − 0.220·21-s + 0.623·22-s + 0.880·23-s − 0.559·24-s − 0.443·25-s − 0.159·26-s + 0.968·27-s − 0.253·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(46.8682\)
Root analytic conductor: \(6.84603\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 91,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 2.40e3T \)
13 \( 1 - 2.85e4T \)
good2 \( 1 + 13.0T + 512T^{2} \)
3 \( 1 + 81.8T + 1.96e4T^{2} \)
5 \( 1 + 1.04e3T + 1.95e6T^{2} \)
11 \( 1 + 5.26e4T + 2.35e9T^{2} \)
17 \( 1 - 6.16e5T + 1.18e11T^{2} \)
19 \( 1 - 2.62e5T + 3.22e11T^{2} \)
23 \( 1 - 1.18e6T + 1.80e12T^{2} \)
29 \( 1 + 3.03e6T + 1.45e13T^{2} \)
31 \( 1 - 5.18e6T + 2.64e13T^{2} \)
37 \( 1 - 8.57e6T + 1.29e14T^{2} \)
41 \( 1 - 2.51e7T + 3.27e14T^{2} \)
43 \( 1 + 2.75e7T + 5.02e14T^{2} \)
47 \( 1 + 4.94e6T + 1.11e15T^{2} \)
53 \( 1 + 1.01e8T + 3.29e15T^{2} \)
59 \( 1 + 1.47e8T + 8.66e15T^{2} \)
61 \( 1 - 8.03e7T + 1.16e16T^{2} \)
67 \( 1 + 5.89e7T + 2.72e16T^{2} \)
71 \( 1 - 8.65e7T + 4.58e16T^{2} \)
73 \( 1 + 3.16e8T + 5.88e16T^{2} \)
79 \( 1 - 3.92e8T + 1.19e17T^{2} \)
83 \( 1 + 8.65e6T + 1.86e17T^{2} \)
89 \( 1 - 1.81e8T + 3.50e17T^{2} \)
97 \( 1 + 1.21e9T + 7.60e17T^{2} \)
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

−11.54537050195680271599124345737, −10.66106167157846318508094053285, −9.541840060206565620049905072609, −8.164132943392254278693835493458, −7.66807921201953469912314058909, −5.69771310254158983522452967443, −4.76174039134577442647949392134, −3.21733715486111751711618091780, −1.04925899222704560976324231974, 0, 1.04925899222704560976324231974, 3.21733715486111751711618091780, 4.76174039134577442647949392134, 5.69771310254158983522452967443, 7.66807921201953469912314058909, 8.164132943392254278693835493458, 9.541840060206565620049905072609, 10.66106167157846318508094053285, 11.54537050195680271599124345737

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