Properties

Label 2-91-91.90-c6-0-23
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $20.9349$
Root an. cond. $4.57546$
Motivic weight $6$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 64·4-s − 198·5-s − 343·7-s + 729·9-s + 2.19e3·13-s + 4.09e3·16-s + 1.14e4·19-s − 1.26e4·20-s − 1.97e4·23-s + 2.35e4·25-s − 2.19e4·28-s + 4.73e4·29-s + 7.81e3·31-s + 6.79e4·35-s + 4.66e4·36-s + 1.24e5·41-s + 2.76e4·43-s − 1.44e5·45-s + 5.34e3·47-s + 1.17e5·49-s + 1.40e5·52-s − 1.23e5·53-s − 2.10e5·59-s − 2.50e5·63-s + 2.62e5·64-s − 4.35e5·65-s + 4.39e5·73-s + ⋯
L(s)  = 1  + 4-s − 1.58·5-s − 7-s + 9-s + 13-s + 16-s + 1.66·19-s − 1.58·20-s − 1.61·23-s + 1.50·25-s − 28-s + 1.94·29-s + 0.262·31-s + 1.58·35-s + 36-s + 1.80·41-s + 0.347·43-s − 1.58·45-s + 0.0514·47-s + 49-s + 52-s − 0.826·53-s − 1.02·59-s − 63-s + 64-s − 1.58·65-s + 1.12·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(20.9349\)
Root analytic conductor: \(4.57546\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: $\chi_{91} (90, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.834983676\)
\(L(\frac12)\) \(\approx\) \(1.834983676\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + p^{3} T \)
13 \( 1 - p^{3} T \)
good2 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
3 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
5 \( 1 + 198 T + p^{6} T^{2} \)
11 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
17 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
19 \( 1 - 11450 T + p^{6} T^{2} \)
23 \( 1 + 19710 T + p^{6} T^{2} \)
29 \( 1 - 47322 T + p^{6} T^{2} \)
31 \( 1 - 7810 T + p^{6} T^{2} \)
37 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
41 \( 1 - 124290 T + p^{6} T^{2} \)
43 \( 1 - 27610 T + p^{6} T^{2} \)
47 \( 1 - 5346 T + p^{6} T^{2} \)
53 \( 1 + 123030 T + p^{6} T^{2} \)
59 \( 1 + 210870 T + p^{6} T^{2} \)
61 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
67 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
71 \( ( 1 - p^{3} T )( 1 + p^{3} T ) \)
73 \( 1 - 439234 T + p^{6} T^{2} \)
79 \( 1 + 953678 T + p^{6} T^{2} \)
83 \( 1 - 733626 T + p^{6} T^{2} \)
89 \( 1 + 571230 T + p^{6} T^{2} \)
97 \( 1 + 1449646 T + p^{6} T^{2} \)
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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

−12.49441924924866274103486417845, −11.94742534616199455018316934349, −10.87390212793482459572903981220, −9.786326236716095479402582486778, −8.077656031098149842566433419297, −7.25284868518562052705655619608, −6.19511514927744308186693114277, −4.11160659142009205949268565211, −3.09174731571012501073196587747, −0.953271029003756203305006493149, 0.953271029003756203305006493149, 3.09174731571012501073196587747, 4.11160659142009205949268565211, 6.19511514927744308186693114277, 7.25284868518562052705655619608, 8.077656031098149842566433419297, 9.786326236716095479402582486778, 10.87390212793482459572903981220, 11.94742534616199455018316934349, 12.49441924924866274103486417845

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