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Properties

Label 2-1040-13.4-c1-0-15
Degree $2$
Conductor $1040$
Sign $0.892 + 0.450i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (0.647 − 1.12i)3-s − i·5-s + (1.06 − 0.613i)7-s + (0.660 + 1.14i)9-s + (2.37 + 1.37i)11-s + (0.155 + 3.60i)13-s + (−1.12 − 0.647i)15-s + (2.79 + 4.83i)17-s + (4.07 − 2.35i)19-s − 1.59i·21-s + (0.674 − 1.16i)23-s − 25-s + 5.59·27-s + (−0.168 + 0.292i)29-s − 2.92i·31-s + ⋯
L(s)  = 1  + (0.374 − 0.647i)3-s − 0.447i·5-s + (0.401 − 0.231i)7-s + (0.220 + 0.381i)9-s + (0.716 + 0.413i)11-s + (0.0431 + 0.999i)13-s + (−0.289 − 0.167i)15-s + (0.677 + 1.17i)17-s + (0.935 − 0.540i)19-s − 0.347i·21-s + (0.140 − 0.243i)23-s − 0.200·25-s + 1.07·27-s + (−0.0313 + 0.0543i)29-s − 0.526i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.892 + 0.450i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.892 + 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.116693181\)
\(L(\frac12)\) \(\approx\) \(2.116693181\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (-0.155 - 3.60i)T \)
good3 \( 1 + (-0.647 + 1.12i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.06 + 0.613i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.37 - 1.37i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.79 - 4.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.07 + 2.35i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.674 + 1.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.168 - 0.292i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.92iT - 31T^{2} \)
37 \( 1 + (1.53 + 0.884i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.36 + 4.82i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.71 + 6.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.95iT - 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (5.10 - 2.94i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.82 + 8.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.58 - 5.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.97 - 2.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.45iT - 73T^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + 1.01iT - 83T^{2} \)
89 \( 1 + (-9.60 - 5.54i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.75 + 2.16i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.810766368744374854194521000863, −8.855958380833381520963238686411, −8.255798231445372388863188873968, −7.29223465993202561265550346461, −6.78054548622656622545916910127, −5.53030988956018997117761899454, −4.56341969029600745041307679926, −3.65856502980039504652473003143, −2.08232664344585830615159807868, −1.30377955131180550174603021177, 1.20808483565098098752016492110, 3.03049418461065005052838936646, 3.46968462063143047423069545208, 4.75094381065492844658055674539, 5.59862675492087456254941864887, 6.63851600983579817868125428407, 7.57271711429644740320510557408, 8.396075323137252788111455415184, 9.314426732344896715274691877987, 9.900138597971014796974109812103

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