Properties

Degree 2
Conductor 73
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 2·5-s + 2·7-s − 3·8-s − 3·9-s + 2·10-s − 2·11-s − 6·13-s + 2·14-s − 16-s + 2·17-s − 3·18-s + 8·19-s − 2·20-s − 2·22-s + 4·23-s − 25-s − 6·26-s − 2·28-s + 2·29-s − 2·31-s + 5·32-s + 2·34-s + 4·35-s + 3·36-s − 6·37-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 1.06·8-s − 9-s + 0.632·10-s − 0.603·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.83·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.377·28-s + 0.371·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s + 0.676·35-s + 1/2·36-s − 0.986·37-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{73} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 73,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.182660467$
$L(\frac12)$  $\approx$  $1.182660467$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 73$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 73$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad73 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.78071511778486, −18.37843267445360, −17.63067364922466, −16.93644286863809, −15.25152910789263, −14.20095311456822, −13.90079370363007, −12.55583978793064, −11.60505678784584, −10.02750085781247, −9.047241577107533, −7.591744378983860, −5.588364505746148, −5.030533566858957, −2.851725941271704, 2.851725941271704, 5.030533566858957, 5.588364505746148, 7.591744378983860, 9.047241577107533, 10.02750085781247, 11.60505678784584, 12.55583978793064, 13.90079370363007, 14.20095311456822, 15.25152910789263, 16.93644286863809, 17.63067364922466, 18.37843267445360, 19.78071511778486

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