Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 $
Sign $0.447 - 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·3-s + (2 + i)5-s + 2i·7-s − 9-s − 2·11-s + 6i·13-s + (1 − 2i)15-s + 2i·17-s + 2·21-s + 4i·23-s + (3 + 4i)25-s + i·27-s − 8·31-s + 2i·33-s + (−2 + 4i)35-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.894 + 0.447i)5-s + 0.755i·7-s − 0.333·9-s − 0.603·11-s + 1.66i·13-s + (0.258 − 0.516i)15-s + 0.485i·17-s + 0.436·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192i·27-s − 1.43·31-s + 0.348i·33-s + (−0.338 + 0.676i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $0.447 - 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{960} (769, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 960,\ (\ :1/2),\ 0.447 - 0.894i)\)
\(L(1)\)  \(\approx\)  \(1.33864 + 0.827325i\)
\(L(\frac12)\)  \(\approx\)  \(1.33864 + 0.827325i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (-2 - i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.12096110965081070063231957866, −9.212099851975808375776571483064, −8.715158220990760030641822257168, −7.46062972812106738853262156018, −6.78755993818593172158689370167, −5.88131568020044699660890229835, −5.24833189380140533966738785239, −3.77205334529209662907691509218, −2.41364340619497992275745377862, −1.76623382470639652667722352824, 0.72102067387008897618786772945, 2.43558900516617993360117622150, 3.50680404998244847127583535008, 4.77142167373993736883932077938, 5.37964832598748357281358339416, 6.27774383454896964527410242383, 7.48323700282678448355579266359, 8.254969611591929950200229108632, 9.193411808798071220306016277263, 10.00753669308419175057550088676

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