Properties

Label 2-91-1.1-c9-0-52
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 27.7·2-s + 192.·3-s + 259.·4-s − 1.97e3·5-s + 5.35e3·6-s + 2.40e3·7-s − 7.01e3·8-s + 1.75e4·9-s − 5.49e4·10-s − 6.44e4·11-s + 5.00e4·12-s + 2.85e4·13-s + 6.66e4·14-s − 3.81e5·15-s − 3.27e5·16-s − 7.36e4·17-s + 4.87e5·18-s + 4.78e5·19-s − 5.13e5·20-s + 4.63e5·21-s − 1.79e6·22-s − 1.93e6·23-s − 1.35e6·24-s + 1.96e6·25-s + 7.93e5·26-s − 4.13e5·27-s + 6.23e5·28-s + ⋯
L(s)  = 1  + 1.22·2-s + 1.37·3-s + 0.506·4-s − 1.41·5-s + 1.68·6-s + 0.377·7-s − 0.605·8-s + 0.891·9-s − 1.73·10-s − 1.32·11-s + 0.697·12-s + 0.277·13-s + 0.463·14-s − 1.94·15-s − 1.24·16-s − 0.213·17-s + 1.09·18-s + 0.842·19-s − 0.717·20-s + 0.519·21-s − 1.63·22-s − 1.44·23-s − 0.832·24-s + 1.00·25-s + 0.340·26-s − 0.149·27-s + 0.191·28-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 \( 1 - 27.7T + 512T^{2} \)
3 \( 1 - 192.T + 1.96e4T^{2} \)
5 \( 1 + 1.97e3T + 1.95e6T^{2} \)
11 \( 1 + 6.44e4T + 2.35e9T^{2} \)
17 \( 1 + 7.36e4T + 1.18e11T^{2} \)
19 \( 1 - 4.78e5T + 3.22e11T^{2} \)
23 \( 1 + 1.93e6T + 1.80e12T^{2} \)
29 \( 1 + 4.36e6T + 1.45e13T^{2} \)
31 \( 1 + 3.84e6T + 2.64e13T^{2} \)
37 \( 1 + 9.08e6T + 1.29e14T^{2} \)
41 \( 1 - 2.19e7T + 3.27e14T^{2} \)
43 \( 1 - 2.54e7T + 5.02e14T^{2} \)
47 \( 1 - 6.57e6T + 1.11e15T^{2} \)
53 \( 1 - 5.07e6T + 3.29e15T^{2} \)
59 \( 1 + 4.39e6T + 8.66e15T^{2} \)
61 \( 1 + 8.31e7T + 1.16e16T^{2} \)
67 \( 1 - 1.50e8T + 2.72e16T^{2} \)
71 \( 1 - 4.39e7T + 4.58e16T^{2} \)
73 \( 1 - 2.79e7T + 5.88e16T^{2} \)
79 \( 1 - 3.99e8T + 1.19e17T^{2} \)
83 \( 1 + 6.07e8T + 1.86e17T^{2} \)
89 \( 1 - 1.00e9T + 3.50e17T^{2} \)
97 \( 1 + 6.96e8T + 7.60e17T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.05842209319255636056734765079, −10.97073232228727492953436032917, −9.248742879019446993506412689529, −8.097920722317711918552529244712, −7.49556640301794637512032788026, −5.50155986518860489600902685326, −4.16357146227950475674860914482, −3.46504364421280210551555014798, −2.36117377561303802496071297074, 0, 2.36117377561303802496071297074, 3.46504364421280210551555014798, 4.16357146227950475674860914482, 5.50155986518860489600902685326, 7.49556640301794637512032788026, 8.097920722317711918552529244712, 9.248742879019446993506412689529, 10.97073232228727492953436032917, 12.05842209319255636056734765079

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