Properties

Degree 2
Conductor $ 2^{8} \cdot 3 \cdot 5 $
Sign $0.316 + 0.948i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
\( \varepsilon \)  =  $0.316 + 0.948i$
motivic weight  =  \(1\)
character  :  $\chi_{3840} (2689, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3840,\ (\ :1/2),\ 0.316 + 0.948i)\)
\(L(1)\)  \(\approx\)  \(1.112749778\)
\(L(\frac12)\)  \(\approx\)  \(1.112749778\)
\(L(\frac{3}{2})\)   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 \( 1 + T \)
5 \( 1 + (1 - 2i)T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
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

−8.325806178665803861551864609891, −7.46483876337736422533928003126, −6.84720043549624350358626348772, −6.14007842572282967512468526645, −5.60358478788117515773885525188, −4.25972264733339860129855668421, −3.82186515933564960511372617941, −2.99222608237856889134273527132, −1.60187568340872147871724595077, −0.42137465715308899645928942561, 1.03975617128410289720254036218, 1.93756372283623915499700052105, 3.35154896961385924792478773281, 4.11548539221463832232546862790, 4.97146855756344697124598267090, 5.62852528967901714807388011888, 6.22258883538268339401687359678, 7.23987004492467640026617087566, 7.908853449102611121118775583871, 8.734379806350938919729633373185

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