Properties

Label 2-91-1.1-c7-0-9
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  − 9.77·2-s − 73.6·3-s − 32.4·4-s + 542.·5-s + 719.·6-s + 343·7-s + 1.56e3·8-s + 3.23e3·9-s − 5.29e3·10-s − 5.39e3·11-s + 2.39e3·12-s + 2.19e3·13-s − 3.35e3·14-s − 3.99e4·15-s − 1.11e4·16-s + 1.84e4·17-s − 3.16e4·18-s − 4.42e4·19-s − 1.76e4·20-s − 2.52e4·21-s + 5.27e4·22-s + 4.38e4·23-s − 1.15e5·24-s + 2.15e5·25-s − 2.14e4·26-s − 7.74e4·27-s − 1.11e4·28-s + ⋯
L(s)  = 1  − 0.863·2-s − 1.57·3-s − 0.253·4-s + 1.93·5-s + 1.36·6-s + 0.377·7-s + 1.08·8-s + 1.48·9-s − 1.67·10-s − 1.22·11-s + 0.399·12-s + 0.277·13-s − 0.326·14-s − 3.05·15-s − 0.681·16-s + 0.908·17-s − 1.27·18-s − 1.48·19-s − 0.492·20-s − 0.595·21-s + 1.05·22-s + 0.750·23-s − 1.70·24-s + 2.76·25-s − 0.239·26-s − 0.757·27-s − 0.0959·28-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

\(L(4)\) \(\approx\) \(0.8245213885\)
\(L(\frac12)\) \(\approx\) \(0.8245213885\)
\(L(\frac{9}{2})\) 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 \( 1 + 9.77T + 128T^{2} \)
3 \( 1 + 73.6T + 2.18e3T^{2} \)
5 \( 1 - 542.T + 7.81e4T^{2} \)
11 \( 1 + 5.39e3T + 1.94e7T^{2} \)
17 \( 1 - 1.84e4T + 4.10e8T^{2} \)
19 \( 1 + 4.42e4T + 8.93e8T^{2} \)
23 \( 1 - 4.38e4T + 3.40e9T^{2} \)
29 \( 1 + 1.34e5T + 1.72e10T^{2} \)
31 \( 1 - 4.98e4T + 2.75e10T^{2} \)
37 \( 1 - 3.29e5T + 9.49e10T^{2} \)
41 \( 1 + 5.50e5T + 1.94e11T^{2} \)
43 \( 1 + 2.95e5T + 2.71e11T^{2} \)
47 \( 1 + 9.95e5T + 5.06e11T^{2} \)
53 \( 1 - 7.55e5T + 1.17e12T^{2} \)
59 \( 1 - 2.25e6T + 2.48e12T^{2} \)
61 \( 1 - 2.50e6T + 3.14e12T^{2} \)
67 \( 1 - 2.12e6T + 6.06e12T^{2} \)
71 \( 1 - 6.30e5T + 9.09e12T^{2} \)
73 \( 1 + 9.58e5T + 1.10e13T^{2} \)
79 \( 1 - 7.53e6T + 1.92e13T^{2} \)
83 \( 1 + 8.33e5T + 2.71e13T^{2} \)
89 \( 1 - 4.73e4T + 4.42e13T^{2} \)
97 \( 1 + 8.93e6T + 8.07e13T^{2} \)
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

−12.88331915110836682219309260194, −11.11166414410223734205282508360, −10.35811280442932356797199148826, −9.777590740314024995194578313053, −8.390266298286211418454182749355, −6.74605590007369034408582659169, −5.57513618811131016258180207116, −4.95643115112012180194369415639, −1.93617478239699795212098804694, −0.73285204090141573717617691639, 0.73285204090141573717617691639, 1.93617478239699795212098804694, 4.95643115112012180194369415639, 5.57513618811131016258180207116, 6.74605590007369034408582659169, 8.390266298286211418454182749355, 9.777590740314024995194578313053, 10.35811280442932356797199148826, 11.11166414410223734205282508360, 12.88331915110836682219309260194

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