↑

Properties

Label 2-538-269.82-c2-0-35
Degree $2$
Conductor $538$
Sign $-0.999 + 0.0118i$
Analytic cond. $14.6594$
Root an. cond. $3.82876$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−2.30 + 2.30i)3-s − 2i·4-s − 2.14·5-s + 4.60i·6-s + (2.52 + 2.52i)7-s + (−2 − 2i)8-s − 1.59i·9-s + (−2.14 + 2.14i)10-s − 2.21i·11-s + (4.60 + 4.60i)12-s − 2.74i·13-s + 5.05·14-s + (4.94 − 4.94i)15-s − 4·16-s + (2.24 − 2.24i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.767 + 0.767i)3-s − 0.5i·4-s − 0.429·5-s + 0.767i·6-s + (0.361 + 0.361i)7-s + (−0.250 − 0.250i)8-s − 0.177i·9-s + (−0.214 + 0.214i)10-s − 0.201i·11-s + (0.383 + 0.383i)12-s − 0.210i·13-s + 0.361·14-s + (0.329 − 0.329i)15-s − 0.250·16-s + (0.132 − 0.132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0118i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.999 + 0.0118i$
Analytic conductor: \(14.6594\)
Root analytic conductor: \(3.82876\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1),\ -0.999 + 0.0118i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.004353585074\)
\(L(\frac12)\) \(\approx\) \(0.004353585074\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
269 \( 1 + (115. - 242. i)T \)
good3 \( 1 + (2.30 - 2.30i)T - 9iT^{2} \)
5 \( 1 + 2.14T + 25T^{2} \)
7 \( 1 + (-2.52 - 2.52i)T + 49iT^{2} \)
11 \( 1 + 2.21iT - 121T^{2} \)
13 \( 1 + 2.74iT - 169T^{2} \)
17 \( 1 + (-2.24 + 2.24i)T - 289iT^{2} \)
19 \( 1 + (-5.98 - 5.98i)T + 361iT^{2} \)
23 \( 1 + 40.7T + 529T^{2} \)
29 \( 1 + (-14.6 - 14.6i)T + 841iT^{2} \)
31 \( 1 + (34.6 + 34.6i)T + 961iT^{2} \)
37 \( 1 + 27.4T + 1.36e3T^{2} \)
41 \( 1 + 52.0T + 1.68e3T^{2} \)
43 \( 1 + 11.6iT - 1.84e3T^{2} \)
47 \( 1 - 78.2T + 2.20e3T^{2} \)
53 \( 1 + 46.8T + 2.80e3T^{2} \)
59 \( 1 + (2.55 + 2.55i)T + 3.48e3iT^{2} \)
61 \( 1 + 119.T + 3.72e3T^{2} \)
67 \( 1 + 39.4T + 4.48e3T^{2} \)
71 \( 1 + (-65.5 - 65.5i)T + 5.04e3iT^{2} \)
73 \( 1 - 120. iT - 5.32e3T^{2} \)
79 \( 1 + 88.1iT - 6.24e3T^{2} \)
83 \( 1 + (62.3 + 62.3i)T + 6.88e3iT^{2} \)
89 \( 1 - 148. iT - 7.92e3T^{2} \)
97 \( 1 + 123. iT - 9.40e3T^{2} \)
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

−10.34676026597107369838223275765, −9.685626222488212526855042875618, −8.460611206874080416292840862681, −7.48404692071838244745815665762, −5.97927201082936551422145685012, −5.42089230943948679633330849442, −4.39496402200771575615143493409, −3.56573755741114894672715128531, −1.98725345752526241372921789163, −0.00152627582434410682874317988, 1.70781447583472920482709055173, 3.53710432822429412241868579446, 4.57240120700939093295624303783, 5.67966871954160980832969638365, 6.46015808849992952054269736633, 7.37152249484753278035915366404, 7.942137005841148999994268194767, 9.140775213255892281481656461686, 10.38963669103792015552293342460, 11.34395283735793150962164896436

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