Properties

Degree 2
Conductor $ 7 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·4-s − 3·5-s + 7-s + 9-s + 4·12-s + 13-s + 6·15-s + 4·16-s − 6·17-s − 7·19-s + 6·20-s − 2·21-s + 3·23-s + 4·25-s + 4·27-s − 2·28-s − 9·29-s + 5·31-s − 3·35-s − 2·36-s + 2·37-s − 2·39-s − 6·41-s − 43-s − 3·45-s + 3·47-s + ⋯
L(s)  = 1  − 1.15·3-s − 4-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 0.277·13-s + 1.54·15-s + 16-s − 1.45·17-s − 1.60·19-s + 1.34·20-s − 0.436·21-s + 0.625·23-s + 4/5·25-s + 0.769·27-s − 0.377·28-s − 1.67·29-s + 0.898·31-s − 0.507·35-s − 1/3·36-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(91\)    =    \(7 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{91} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 91,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.18950628174769, −18.46578143001059, −17.40200196338876, −16.87342386375902, −15.58158080137234, −14.79958029796450, −13.32859477024007, −12.40148043503927, −11.34343268397352, −10.73267410274711, −8.965474064429713, −8.023034081311284, −6.484321814193631, −4.954540461079239, −4.046096433153055, 0, 4.046096433153055, 4.954540461079239, 6.484321814193631, 8.023034081311284, 8.965474064429713, 10.73267410274711, 11.34343268397352, 12.40148043503927, 13.32859477024007, 14.79958029796450, 15.58158080137234, 16.87342386375902, 17.40200196338876, 18.46578143001059, 19.18950628174769

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