↑

Properties

Label 2-1014-39.8-c1-0-41
Degree $2$
Conductor $1014$
Sign $-0.892 + 0.450i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.739 + 1.56i)3-s − 1.00i·4-s + (2.01 − 2.01i)5-s + (0.584 + 1.63i)6-s + (−2.99 + 2.99i)7-s + (−0.707 − 0.707i)8-s + (−1.90 − 2.31i)9-s − 2.84i·10-s + (−2.96 − 2.96i)11-s + (1.56 + 0.739i)12-s + 4.24i·14-s + (1.66 + 4.64i)15-s − 1.00·16-s − 0.381·17-s + (−2.98 − 0.291i)18-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.427 + 0.904i)3-s − 0.500i·4-s + (0.900 − 0.900i)5-s + (0.238 + 0.665i)6-s + (−1.13 + 1.13i)7-s + (−0.250 − 0.250i)8-s + (−0.635 − 0.772i)9-s − 0.900i·10-s + (−0.893 − 0.893i)11-s + (0.452 + 0.213i)12-s + 1.13i·14-s + (0.429 + 1.19i)15-s − 0.250·16-s − 0.0926·17-s + (−0.703 − 0.0685i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 + 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.892 + 0.450i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ -0.892 + 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6703100688\)
\(L(\frac12)\) \(\approx\) \(0.6703100688\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.739 - 1.56i)T \)
13 \( 1 \)
good5 \( 1 + (-2.01 + 2.01i)T - 5iT^{2} \)
7 \( 1 + (2.99 - 2.99i)T - 7iT^{2} \)
11 \( 1 + (2.96 + 2.96i)T + 11iT^{2} \)
17 \( 1 + 0.381T + 17T^{2} \)
19 \( 1 + (5.03 + 5.03i)T + 19iT^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 - 0.246iT - 29T^{2} \)
31 \( 1 + (4.34 + 4.34i)T + 31iT^{2} \)
37 \( 1 + (-1.78 + 1.78i)T - 37iT^{2} \)
41 \( 1 + (-0.0932 + 0.0932i)T - 41iT^{2} \)
43 \( 1 - 3.51iT - 43T^{2} \)
47 \( 1 + (0.565 + 0.565i)T + 47iT^{2} \)
53 \( 1 + 2.00iT - 53T^{2} \)
59 \( 1 + (-5.80 - 5.80i)T + 59iT^{2} \)
61 \( 1 + 9.90T + 61T^{2} \)
67 \( 1 + (3.97 + 3.97i)T + 67iT^{2} \)
71 \( 1 + (-7.74 + 7.74i)T - 71iT^{2} \)
73 \( 1 + (-8.12 + 8.12i)T - 73iT^{2} \)
79 \( 1 + 5.47T + 79T^{2} \)
83 \( 1 + (12.8 - 12.8i)T - 83iT^{2} \)
89 \( 1 + (6.79 + 6.79i)T + 89iT^{2} \)
97 \( 1 + (-2.90 - 2.90i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.492792633397850731223659289288, −9.195097005440511430797789639503, −8.407984826546100250199314773732, −6.49176686693053139185595852501, −5.80233539509229761231162527808, −5.36527076205159375133273936470, −4.39730486488507540900346736954, −3.15818061373971692832712870758, −2.30909593127113053957792222402, −0.24459406486980114567072154278, 1.94756541475491763277223821609, 2.96785017262642802512590302890, 4.17662493174868808419636194057, 5.48169709688468407923548019025, 6.24563463014002901275718094714, 6.84615149545185225012965684796, 7.36969256973934135345727085470, 8.305882313218842665189374795633, 9.766381101414611068696970568329, 10.36322046560501606811497492670

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