Properties

Degree 2
Conductor $ 2^{5} \cdot 3 $
Sign $0.816 + 0.577i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.41i)3-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 5.65i·17-s − 2·19-s − 5·25-s + (−5.00 − 1.41i)27-s + (4.00 + 2.82i)33-s − 11.3i·41-s + 10·43-s + 7·49-s + (8.00 + 5.65i)51-s + (−2 + 2.82i)57-s − 14.1i·59-s − 14·67-s + ⋯
L(s)  = 1  + (0.577 − 0.816i)3-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 1.37i·17-s − 0.458·19-s − 25-s + (−0.962 − 0.272i)27-s + (0.696 + 0.492i)33-s − 1.76i·41-s + 1.52·43-s + 49-s + (1.12 + 0.792i)51-s + (−0.264 + 0.374i)57-s − 1.84i·59-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(96\)    =    \(2^{5} \cdot 3\)
\( \varepsilon \)  =  $0.816 + 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{96} (47, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 96,\ (\ :1/2),\ 0.816 + 0.577i)$
$L(1)$  $\approx$  $1.08458 - 0.344722i$
$L(\frac12)$  $\approx$  $1.08458 - 0.344722i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1 + 1.41i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−13.83052280484096635574528108503, −12.77691477777785881349281164349, −12.10706443513036903495494494042, −10.63625573446747533218639610596, −9.349238772081262739397023984498, −8.221852677201233064086912461177, −7.18846721052102894550229109732, −5.94979681928227901041780787872, −3.94892189051276045666934919575, −2.05544024252426502892033942734, 2.85796536911498613400902789739, 4.35093785569216056385832285131, 5.77683474317835016152986578228, 7.55043553750843323511328883538, 8.727473141562628758708798405880, 9.656089035790711538682577334756, 10.79247496690339236535851258895, 11.76916993795214137709269398896, 13.36478006518469625335286818462, 14.06394328773157206212895907659

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