Properties

Label 2-2240-28.27-c1-0-63
Degree $2$
Conductor $2240$
Sign $-0.954 + 0.299i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.792·3-s i·5-s + (−0.792 − 2.52i)7-s − 2.37·9-s − 0.792i·11-s − 5.37i·13-s − 0.792i·15-s + 3.37i·17-s + 3.46·19-s + (−0.627 − 2i)21-s + 1.87i·23-s − 25-s − 4.25·27-s + 5.37·29-s − 8.51·31-s + ⋯
L(s)  = 1  + 0.457·3-s − 0.447i·5-s + (−0.299 − 0.954i)7-s − 0.790·9-s − 0.238i·11-s − 1.49i·13-s − 0.204i·15-s + 0.817i·17-s + 0.794·19-s + (−0.136 − 0.436i)21-s + 0.391i·23-s − 0.200·25-s − 0.819·27-s + 0.997·29-s − 1.52·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.954 + 0.299i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.954 + 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9241154039\)
\(L(\frac12)\) \(\approx\) \(0.9241154039\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (0.792 + 2.52i)T \)
good3 \( 1 - 0.792T + 3T^{2} \)
11 \( 1 + 0.792iT - 11T^{2} \)
13 \( 1 + 5.37iT - 13T^{2} \)
17 \( 1 - 3.37iT - 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 - 1.87iT - 23T^{2} \)
29 \( 1 - 5.37T + 29T^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 - 0.744T + 37T^{2} \)
41 \( 1 - 2.74iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 + 6.63T + 59T^{2} \)
61 \( 1 + 0.744iT - 61T^{2} \)
67 \( 1 + 6.63iT - 67T^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 - 2.74iT - 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 17.4iT - 89T^{2} \)
97 \( 1 + 2.11iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.553061272697279386793479902799, −7.976790226022694639906757624429, −7.38994900037063348336550377348, −6.23594412877000822938054349039, −5.55917740040895117090758296745, −4.66352965057292916225887749518, −3.44373831069238384260059347159, −3.12097510560604680922931731132, −1.55687443235877722866599115349, −0.28394487742746485842902385082, 1.83047091297745045650366766625, 2.72439388245439149759730316225, 3.39723564419530370096073152584, 4.60890365596856585522608413418, 5.46936533657421645619196143308, 6.35730226838477357418813328199, 6.99444541656982700494895027335, 7.927030017724001672771358577304, 8.731318321033163417965895055019, 9.378176910970264200767337609165

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