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Properties

Label 2-2240-140.139-c1-0-59
Degree $2$
Conductor $2240$
Sign $-0.106 + 0.994i$
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  − 2.40i·3-s + (0.475 − 2.18i)5-s + (0.283 + 2.63i)7-s − 2.79·9-s + 3.03i·11-s + 5.37·13-s + (−5.26 − 1.14i)15-s − 3.16·17-s + 3.51·19-s + (6.33 − 0.682i)21-s + 4.66·23-s + (−4.54 − 2.07i)25-s − 0.492i·27-s − 0.705·29-s + 9.63·31-s + ⋯
L(s)  = 1  − 1.38i·3-s + (0.212 − 0.977i)5-s + (0.107 + 0.994i)7-s − 0.931·9-s + 0.915i·11-s + 1.48·13-s + (−1.35 − 0.295i)15-s − 0.768·17-s + 0.806·19-s + (1.38 − 0.148i)21-s + 0.972·23-s + (−0.909 − 0.415i)25-s − 0.0947i·27-s − 0.130·29-s + 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.106 + 0.994i$
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: \((2,\ 2240,\ (\ :1/2),\ -0.106 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.049389428\)
\(L(\frac12)\) \(\approx\) \(2.049389428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.475 + 2.18i)T \)
7 \( 1 + (-0.283 - 2.63i)T \)
good3 \( 1 + 2.40iT - 3T^{2} \)
11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 - 5.37T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 - 4.66T + 23T^{2} \)
29 \( 1 + 0.705T + 29T^{2} \)
31 \( 1 - 9.63T + 31T^{2} \)
37 \( 1 + 8.76iT - 37T^{2} \)
41 \( 1 + 4.19iT - 41T^{2} \)
43 \( 1 + 5.55T + 43T^{2} \)
47 \( 1 + 5.97iT - 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 - 7.34T + 67T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 1.88T + 73T^{2} \)
79 \( 1 - 0.664iT - 79T^{2} \)
83 \( 1 + 0.0137iT - 83T^{2} \)
89 \( 1 - 5.01iT - 89T^{2} \)
97 \( 1 + 3.16T + 97T^{2} \)
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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

−8.557551255034956597266048760467, −8.292432684573013804833554613717, −7.18978535517295148223372620749, −6.57702761636051322106164960230, −5.72650558693346025890108615482, −5.08024309240439863896606423125, −3.96749809684666206608000420220, −2.53515920739144467319815477001, −1.76248672567953818285493821809, −0.870321811261729817228355753696, 1.15850719382582534132776888329, 3.01895869209771292677832599073, 3.44260305530913122362506814853, 4.30528333620971540390309607083, 5.10484500379399814005842300649, 6.23299546755654538961904522362, 6.67153746143800796253773904504, 7.82429882439932462920662804822, 8.577501028173135474349850967547, 9.410645932761622804019047640447

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