Properties

Degree 2
Conductor $ 2^{3} \cdot 19 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·7-s − 2·9-s + 2·11-s + 13-s − 5·17-s + 19-s + 3·21-s − 23-s − 5·25-s − 5·27-s − 3·29-s + 4·31-s + 2·33-s + 2·37-s + 39-s − 8·41-s − 8·43-s − 8·47-s + 2·49-s − 5·51-s + 9·53-s + 57-s + 59-s + 14·61-s − 6·63-s + 13·67-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s − 1.21·17-s + 0.229·19-s + 0.654·21-s − 0.208·23-s − 25-s − 0.962·27-s − 0.557·29-s + 0.718·31-s + 0.348·33-s + 0.328·37-s + 0.160·39-s − 1.24·41-s − 1.21·43-s − 1.16·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s + 0.132·57-s + 0.130·59-s + 1.79·61-s − 0.755·63-s + 1.58·67-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(152\)    =    \(2^{3} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{152} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 152,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.367065214$
$L(\frac12)$  $\approx$  $1.367065214$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−19.62350267701707, −18.41296381355187, −17.59773361077071, −16.89305372734120, −15.56647822719036, −14.79609169805219, −14.01268976830858, −13.25663875750955, −11.66341543594217, −11.29580326541951, −9.815161388517684, −8.649214735442594, −8.058703950373763, −6.615707517471348, −5.189151750260142, −3.802990804644453, −2.061138052027253, 2.061138052027253, 3.802990804644453, 5.189151750260142, 6.615707517471348, 8.058703950373763, 8.649214735442594, 9.815161388517684, 11.29580326541951, 11.66341543594217, 13.25663875750955, 14.01268976830858, 14.79609169805219, 15.56647822719036, 16.89305372734120, 17.59773361077071, 18.41296381355187, 19.62350267701707

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