Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 $
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 + 8-s + 9-s − 12-s − 2·13-s + 16-s + 18-s − 23-s − 24-s − 25-s − 2·26-s − 27-s + 32-s + 36-s + 2·39-s − 46-s + 2·47-s − 48-s − 49-s − 50-s − 2·52-s − 54-s + 2·59-s + 64-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 2·13-s + 16-s + 18-s − 23-s − 24-s − 25-s − 2·26-s − 27-s + 32-s + 36-s + 2·39-s − 46-s + 2·47-s − 48-s − 49-s − 50-s − 2·52-s − 54-s + 2·59-s + 64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(276\)    =    \(2^{2} \cdot 3 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{276} (275, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 276,\ (\ :0),\ 1)$
$L(\frac{1}{2})$  $\approx$  $0.9935342802$
$L(\frac12)$  $\approx$  $0.9935342802$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 + T )^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )^{2} \)
79 \( 1 + T^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
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

−12.10785865755557287761980673259, −11.55009886897249101253129072936, −10.39256687254628662397292127470, −9.729189916338952454796774846704, −7.75485496510642676156134406135, −6.99692796576244713645341749392, −5.88862139374234201961255239624, −5.02813123257204501746520012419, −4.04599332383850795239322337996, −2.24544852622770945663253188417, 2.24544852622770945663253188417, 4.04599332383850795239322337996, 5.02813123257204501746520012419, 5.88862139374234201961255239624, 6.99692796576244713645341749392, 7.75485496510642676156134406135, 9.729189916338952454796774846704, 10.39256687254628662397292127470, 11.55009886897249101253129072936, 12.10785865755557287761980673259

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