Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.249 + 0.968i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 0.118i)2-s + (1.97 − 0.335i)4-s − 3.46i·5-s + 7-s + (2.73 − 0.707i)8-s + (−0.411 − 4.88i)10-s + 3.31i·11-s − 3.10i·13-s + (1.40 − 0.118i)14-s + (3.77 − 1.32i)16-s − 1.40·17-s − 4.80i·19-s + (−1.16 − 6.82i)20-s + (0.394 + 4.67i)22-s + 8.79·23-s + ⋯
L(s)  = 1  + (0.996 − 0.0841i)2-s + (0.985 − 0.167i)4-s − 1.54i·5-s + 0.377·7-s + (0.968 − 0.249i)8-s + (−0.130 − 1.54i)10-s + 1.00i·11-s − 0.862i·13-s + (0.376 − 0.0317i)14-s + (0.943 − 0.330i)16-s − 0.340·17-s − 1.10i·19-s + (−0.259 − 1.52i)20-s + (0.0841 + 0.997i)22-s + 1.83·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.249 + 0.968i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.249 + 0.968i)$
$L(1)$  $\approx$  $3.448280268$
$L(\frac12)$  $\approx$  $3.448280268$
$L(\frac{3}{2})$   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 \( 1 + (-1.40 + 0.118i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 - 3.31iT - 11T^{2} \)
13 \( 1 + 3.10iT - 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
23 \( 1 - 8.79T + 23T^{2} \)
29 \( 1 - 9.87iT - 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 + 5.42iT - 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 + 7.03iT - 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 6.51iT - 53T^{2} \)
59 \( 1 - 3.89iT - 59T^{2} \)
61 \( 1 - 9.42iT - 61T^{2} \)
67 \( 1 - 0.909iT - 67T^{2} \)
71 \( 1 + 6.42T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 0.370iT - 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 - 1.63T + 97T^{2} \)
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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

−9.018013975864045637594091733781, −8.752024258288839323306186227905, −7.36025445287344087466080230790, −7.03966590338183921878234855276, −5.48866851559313780801247317751, −5.11934296508039697280992399236, −4.51140148021007400514079575525, −3.40263916232106973842280445810, −2.11280734128788328152834305664, −1.03338253282637387859277275967, 1.79612924101506294165866867126, 2.88704889539099894936266425209, 3.55090371428091367708515492776, 4.51938150377795619198732921532, 5.68048167930790194842162194235, 6.33197356328618178381304746721, 7.02916564581856971309701845577, 7.71892034159813349584680191664, 8.699118923988865007030270443278, 9.910959780625221594585750855058

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