Properties

Label 2-91-1.1-c7-0-39
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.1·2-s + 6.78·3-s + 73.0·4-s − 4.23·5-s + 96.1·6-s + 343·7-s − 779.·8-s − 2.14e3·9-s − 60.0·10-s − 7.89e3·11-s + 495.·12-s − 2.19e3·13-s + 4.86e3·14-s − 28.7·15-s − 2.03e4·16-s + 3.19e4·17-s − 3.03e4·18-s − 9.14e3·19-s − 309.·20-s + 2.32e3·21-s − 1.11e5·22-s + 2.87e4·23-s − 5.28e3·24-s − 7.81e4·25-s − 3.11e4·26-s − 2.93e4·27-s + 2.50e4·28-s + ⋯
L(s)  = 1  + 1.25·2-s + 0.145·3-s + 0.570·4-s − 0.0151·5-s + 0.181·6-s + 0.377·7-s − 0.538·8-s − 0.978·9-s − 0.0189·10-s − 1.78·11-s + 0.0827·12-s − 0.277·13-s + 0.473·14-s − 0.00219·15-s − 1.24·16-s + 1.57·17-s − 1.22·18-s − 0.305·19-s − 0.00864·20-s + 0.0548·21-s − 2.24·22-s + 0.492·23-s − 0.0780·24-s − 0.999·25-s − 0.347·26-s − 0.287·27-s + 0.215·28-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: \(1\)
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 \( 1 - 14.1T + 128T^{2} \)
3 \( 1 - 6.78T + 2.18e3T^{2} \)
5 \( 1 + 4.23T + 7.81e4T^{2} \)
11 \( 1 + 7.89e3T + 1.94e7T^{2} \)
17 \( 1 - 3.19e4T + 4.10e8T^{2} \)
19 \( 1 + 9.14e3T + 8.93e8T^{2} \)
23 \( 1 - 2.87e4T + 3.40e9T^{2} \)
29 \( 1 + 1.34e4T + 1.72e10T^{2} \)
31 \( 1 + 1.72e5T + 2.75e10T^{2} \)
37 \( 1 + 2.45e5T + 9.49e10T^{2} \)
41 \( 1 + 7.78e5T + 1.94e11T^{2} \)
43 \( 1 - 7.13e5T + 2.71e11T^{2} \)
47 \( 1 - 1.32e5T + 5.06e11T^{2} \)
53 \( 1 - 7.43e5T + 1.17e12T^{2} \)
59 \( 1 - 1.05e6T + 2.48e12T^{2} \)
61 \( 1 - 1.76e6T + 3.14e12T^{2} \)
67 \( 1 - 7.44e5T + 6.06e12T^{2} \)
71 \( 1 - 3.83e5T + 9.09e12T^{2} \)
73 \( 1 + 3.29e6T + 1.10e13T^{2} \)
79 \( 1 + 2.30e6T + 1.92e13T^{2} \)
83 \( 1 + 3.15e6T + 2.71e13T^{2} \)
89 \( 1 + 1.36e6T + 4.42e13T^{2} \)
97 \( 1 + 1.40e6T + 8.07e13T^{2} \)
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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

−12.37941671014015153012380627500, −11.39156553771675113770077514786, −10.15893690910566150287902207524, −8.618618278341873085081815392218, −7.50699501233276600213730785139, −5.66625394574512151500768724962, −5.17409815174947896522100673508, −3.53134722916179190378439356493, −2.44454168611612250213989416158, 0, 2.44454168611612250213989416158, 3.53134722916179190378439356493, 5.17409815174947896522100673508, 5.66625394574512151500768724962, 7.50699501233276600213730785139, 8.618618278341873085081815392218, 10.15893690910566150287902207524, 11.39156553771675113770077514786, 12.37941671014015153012380627500

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