Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 7 $
Sign $0.970 + 0.239i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 + 0.114i)2-s + (1.97 + 0.321i)4-s − 1.12i·5-s + 7-s + (2.74 + 0.678i)8-s + (0.128 − 1.59i)10-s − 4.76i·11-s − 0.456i·13-s + (1.40 + 0.114i)14-s + (3.79 + 1.26i)16-s − 0.415·17-s + 7.63i·19-s + (0.362 − 2.22i)20-s + (0.543 − 6.71i)22-s − 1.58·23-s + ⋯
L(s)  = 1  + (0.996 + 0.0806i)2-s + (0.986 + 0.160i)4-s − 0.504i·5-s + 0.377·7-s + (0.970 + 0.239i)8-s + (0.0407 − 0.503i)10-s − 1.43i·11-s − 0.126i·13-s + (0.376 + 0.0304i)14-s + (0.948 + 0.317i)16-s − 0.100·17-s + 1.75i·19-s + (0.0811 − 0.498i)20-s + (0.115 − 1.43i)22-s − 0.330·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.970 + 0.239i$
motivic weight  =  \(1\)
character  :  $\chi_{504} (253, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 504,\ (\ :1/2),\ 0.970 + 0.239i)$
$L(1)$  $\approx$  $2.72720 - 0.331923i$
$L(\frac12)$  $\approx$  $2.72720 - 0.331923i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.114i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 1.12iT - 5T^{2} \)
11 \( 1 + 4.76iT - 11T^{2} \)
13 \( 1 + 0.456iT - 13T^{2} \)
17 \( 1 + 0.415T + 17T^{2} \)
19 \( 1 - 7.63iT - 19T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 - 6.72iT - 29T^{2} \)
31 \( 1 + 5.89T + 31T^{2} \)
37 \( 1 + 5.89iT - 37T^{2} \)
41 \( 1 - 0.415T + 41T^{2} \)
43 \( 1 + 9.43iT - 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 7.63iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 - 1.80iT - 61T^{2} \)
67 \( 1 - 8.09iT - 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 - 5.53iT - 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−10.99627749051595186608476081206, −10.37137589871760183763488548060, −8.869376384141027822873997772291, −8.144189992303604575116209533101, −7.12999354779534989655671629363, −5.87674141930311228373888966753, −5.37949010843785337960412094676, −4.10103043067736742002546404459, −3.17910039594386657395572919818, −1.54828611847165281916656256033, 1.92146194737866490544514208631, 3.00232858290797988011737880139, 4.42073875787453191274738032493, 5.00570832815294868399610530159, 6.41570332406215325822000921135, 7.04971036118181880435626049908, 7.943384430751275810298933919535, 9.403554078571833659579657016436, 10.26018684829271363477559873193, 11.23171246911377278564778226587

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