Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 12-s + 13-s + 14-s + 16-s − 17-s − 19-s − 21-s − 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s − 32-s + 34-s − 2·37-s + 38-s + 39-s + 42-s + 46-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 12-s + 13-s + 14-s + 16-s − 17-s − 19-s − 21-s − 23-s − 24-s + 25-s − 26-s − 27-s − 28-s + 29-s − 32-s + 34-s − 2·37-s + 38-s + 39-s + 42-s + 46-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(152\)    =    \(2^{3} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{152} (37, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 152,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.5255734463$
$L(\frac12)$  $\approx$  $0.5255734463$
$L(1)$   not available
$L(1)$   not available

Euler product

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

−13.27399541754351592744446478828, −12.20286584791493355118555394928, −10.89692957890453178097024652546, −10.03706376682674974881302024345, −8.724373722575583812286798578832, −8.616744569991860759400358457809, −7.04273464292366857250117884391, −6.10266027944254780694352976077, −3.65604067108832041139853999554, −2.38472062956344778846731955046, 2.38472062956344778846731955046, 3.65604067108832041139853999554, 6.10266027944254780694352976077, 7.04273464292366857250117884391, 8.616744569991860759400358457809, 8.724373722575583812286798578832, 10.03706376682674974881302024345, 10.89692957890453178097024652546, 12.20286584791493355118555394928, 13.27399541754351592744446478828

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