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Properties

Label 2-2240-16.5-c1-0-27
Degree $2$
Conductor $2240$
Sign $0.997 + 0.0682i$
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  + (1.67 − 1.67i)3-s + (0.707 + 0.707i)5-s − i·7-s − 2.64i·9-s + (4.03 + 4.03i)11-s + (−4.75 + 4.75i)13-s + 2.37·15-s + 4.01·17-s + (−1.17 + 1.17i)19-s + (−1.67 − 1.67i)21-s − 0.610i·23-s + 1.00i·25-s + (0.601 + 0.601i)27-s + (2.42 − 2.42i)29-s + 8.63·31-s + ⋯
L(s)  = 1  + (0.969 − 0.969i)3-s + (0.316 + 0.316i)5-s − 0.377i·7-s − 0.880i·9-s + (1.21 + 1.21i)11-s + (−1.31 + 1.31i)13-s + 0.613·15-s + 0.972·17-s + (−0.270 + 0.270i)19-s + (−0.366 − 0.366i)21-s − 0.127i·23-s + 0.200i·25-s + (0.115 + 0.115i)27-s + (0.450 − 0.450i)29-s + 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0682i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (561, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ 0.997 + 0.0682i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.805475175\)
\(L(\frac12)\) \(\approx\) \(2.805475175\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-1.67 + 1.67i)T - 3iT^{2} \)
11 \( 1 + (-4.03 - 4.03i)T + 11iT^{2} \)
13 \( 1 + (4.75 - 4.75i)T - 13iT^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
19 \( 1 + (1.17 - 1.17i)T - 19iT^{2} \)
23 \( 1 + 0.610iT - 23T^{2} \)
29 \( 1 + (-2.42 + 2.42i)T - 29iT^{2} \)
31 \( 1 - 8.63T + 31T^{2} \)
37 \( 1 + (1.14 + 1.14i)T + 37iT^{2} \)
41 \( 1 - 1.24iT - 41T^{2} \)
43 \( 1 + (-8.70 - 8.70i)T + 43iT^{2} \)
47 \( 1 + 7.92T + 47T^{2} \)
53 \( 1 + (5.64 + 5.64i)T + 53iT^{2} \)
59 \( 1 + (-1.23 - 1.23i)T + 59iT^{2} \)
61 \( 1 + (-5.71 + 5.71i)T - 61iT^{2} \)
67 \( 1 + (2.38 - 2.38i)T - 67iT^{2} \)
71 \( 1 + 1.57iT - 71T^{2} \)
73 \( 1 + 1.85iT - 73T^{2} \)
79 \( 1 + 3.14T + 79T^{2} \)
83 \( 1 + (-6.01 + 6.01i)T - 83iT^{2} \)
89 \( 1 + 18.0iT - 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.095261808405993804662866753719, −8.043402125188848021293221859093, −7.49945874272526693871655656222, −6.74833722021806818881881898996, −6.40000316993415399078667654223, −4.82924830381433680282203676882, −4.14591166270688156169300606158, −2.94452625523962188318520486301, −2.08361102424578986714137821503, −1.36314602139127955045451972500, 0.951822063118836118268691745381, 2.60269265294933882926241842862, 3.17224289208144189823386203917, 4.05538658909694475823998823359, 5.01987681336230084803039161178, 5.71326622068964338280785949795, 6.66619130598902219439818053924, 7.86069929621551248476857391373, 8.440624041199107169403425836884, 9.048718740861423662043793415440

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