Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \cdot 7 $
Sign $-0.994 + 0.100i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44i·3-s + (−0.224 − 2.22i)5-s i·7-s − 2.99·9-s − 4.89·11-s − 0.449i·13-s + (−5.44 + 0.550i)15-s − 2i·17-s + 6.44·19-s − 2.44·21-s + 6.89i·23-s + (−4.89 + i)25-s + 2.89·29-s + 0.898·31-s + 11.9i·33-s + ⋯
L(s)  = 1  − 1.41i·3-s + (−0.100 − 0.994i)5-s − 0.377i·7-s − 0.999·9-s − 1.47·11-s − 0.124i·13-s + (−1.40 + 0.142i)15-s − 0.485i·17-s + 1.47·19-s − 0.534·21-s + 1.43i·23-s + (−0.979 + 0.200i)25-s + 0.538·29-s + 0.161·31-s + 2.08i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.994 + 0.100i$
motivic weight  =  \(1\)
character  :  $\chi_{560} (449, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 560,\ (\ :1/2),\ -0.994 + 0.100i)$
$L(1)$  $\approx$  $0.0547101 - 1.08590i$
$L(\frac12)$  $\approx$  $0.0547101 - 1.08590i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.224 + 2.22i)T \)
7 \( 1 + iT \)
good3 \( 1 + 2.44iT - 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 0.449iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 - 6.89iT - 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 0.898T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 8.89iT - 43T^{2} \)
47 \( 1 + 0.898iT - 47T^{2} \)
53 \( 1 + 1.10iT - 53T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 6.89iT - 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−10.30938474347751412150041228464, −9.379112251077911053384122580331, −8.213501166713106520238873691060, −7.68414523634273407074751041424, −6.99415327141256090395497325092, −5.61574650205583044526235799684, −4.99537457163729420102410260742, −3.30188167355794241111823405207, −1.86767772322833534227136792854, −0.61608851251157899507562863377, 2.62398767237310129532409966925, 3.40736505686345329035316448916, 4.65498515269625819021756396253, 5.44633409098404133479948268008, 6.57452826734288802278571971606, 7.76284298523440072796984466160, 8.616238143776493231826296875299, 9.846874011343112458534974745042, 10.19612968375096452621348887364, 10.94233245004098263325515437387

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