Properties

Degree 2
Conductor $ 5 \cdot 37 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s + 5-s − 3·7-s − 2·9-s − 5·11-s + 2·12-s + 4·13-s − 15-s + 4·16-s − 4·17-s − 8·19-s − 2·20-s + 3·21-s + 4·23-s + 25-s + 5·27-s + 6·28-s + 4·29-s + 2·31-s + 5·33-s − 3·35-s + 4·36-s + 37-s − 4·39-s − 5·41-s − 6·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 1.50·11-s + 0.577·12-s + 1.10·13-s − 0.258·15-s + 16-s − 0.970·17-s − 1.83·19-s − 0.447·20-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.962·27-s + 1.13·28-s + 0.742·29-s + 0.359·31-s + 0.870·33-s − 0.507·35-s + 2/3·36-s + 0.164·37-s − 0.640·39-s − 0.780·41-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(185\)    =    \(5 \cdot 37\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{185} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 185,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;37\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;37\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
37 \( 1 - T \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−18.98437678671841, −18.41439993427333, −17.53098723716169, −16.88566575790104, −15.89283651087105, −14.97983668444252, −13.51608145765417, −13.28560563857172, −12.39360295919822, −10.82163125080847, −10.33004705055258, −8.998244863867538, −8.371026417403939, −6.539640057935940, −5.744803669168943, −4.568750928167755, −2.945100704752466, 0, 2.945100704752466, 4.568750928167755, 5.744803669168943, 6.539640057935940, 8.371026417403939, 8.998244863867538, 10.33004705055258, 10.82163125080847, 12.39360295919822, 13.28560563857172, 13.51608145765417, 14.97983668444252, 15.89283651087105, 16.88566575790104, 17.53098723716169, 18.41439993427333, 18.98437678671841

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