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Properties

Label 2-2240-140.139-c1-0-27
Degree $2$
Conductor $2240$
Sign $-0.551 - 0.834i$
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  + 2.13i·3-s + (1.94 + 1.10i)5-s + (−2.35 − 1.19i)7-s − 1.56·9-s + 2.33i·11-s + 1.09·13-s + (−2.35 + 4.15i)15-s + 4.98·17-s + 2.57·19-s + (2.56 − 5.03i)21-s + 6.04·23-s + (2.56 + 4.29i)25-s + 3.07i·27-s − 0.561·29-s − 6.59·31-s + ⋯
L(s)  = 1  + 1.23i·3-s + (0.869 + 0.493i)5-s + (−0.891 − 0.453i)7-s − 0.520·9-s + 0.703i·11-s + 0.302·13-s + (−0.608 + 1.07i)15-s + 1.20·17-s + 0.590·19-s + (0.558 − 1.09i)21-s + 1.25·23-s + (0.512 + 0.858i)25-s + 0.591i·27-s − 0.104·29-s − 1.18·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.551 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.959926022\)
\(L(\frac12)\) \(\approx\) \(1.959926022\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.10i)T \)
7 \( 1 + (2.35 + 1.19i)T \)
good3 \( 1 - 2.13iT - 3T^{2} \)
11 \( 1 - 2.33iT - 11T^{2} \)
13 \( 1 - 1.09T + 13T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 - 6.04T + 23T^{2} \)
29 \( 1 + 0.561T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 - 5.49iT - 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 - 1.32T + 43T^{2} \)
47 \( 1 - 9.74iT - 47T^{2} \)
53 \( 1 - 8.58iT - 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 0.620iT - 61T^{2} \)
67 \( 1 + 4.71T + 67T^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 - 9.96T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−9.511016273849510294735451839597, −9.005925882462392525955297586319, −7.55183484848737823494635233586, −7.01867868268958681380746895256, −6.01870824871310706278717435513, −5.33227553881445596856364044529, −4.46662274605001211618295741222, −3.44554649623722902287236835508, −2.95559126167480570765904769182, −1.40037797212411368421638079676, 0.73172149128843069526677008679, 1.63680747095993844109366835872, 2.73473864981549259605386466017, 3.56803275169950918973784352269, 5.16165200883442685572194149302, 5.75302073939440478049776807207, 6.40849063723967602255874315805, 7.13477438740762940664566098172, 7.956435076476914917568321301498, 8.800821908916012127370165196245

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