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Properties

Label 2-538-269.82-c2-0-17
Degree $2$
Conductor $538$
Sign $0.531 - 0.847i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−2.38 + 2.38i)3-s − 2i·4-s − 4.67·5-s − 4.77i·6-s + (4.02 + 4.02i)7-s + (2 + 2i)8-s − 2.41i·9-s + (4.67 − 4.67i)10-s − 0.758i·11-s + (4.77 + 4.77i)12-s − 22.3i·13-s − 8.04·14-s + (11.1 − 11.1i)15-s − 4·16-s + (7.71 − 7.71i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.796 + 0.796i)3-s − 0.5i·4-s − 0.934·5-s − 0.796i·6-s + (0.574 + 0.574i)7-s + (0.250 + 0.250i)8-s − 0.268i·9-s + (0.467 − 0.467i)10-s − 0.0689i·11-s + (0.398 + 0.398i)12-s − 1.71i·13-s − 0.574·14-s + (0.744 − 0.744i)15-s − 0.250·16-s + (0.453 − 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.531 - 0.847i$
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: \((2,\ 538,\ (\ :1),\ 0.531 - 0.847i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7824168024\)
\(L(\frac12)\) \(\approx\) \(0.7824168024\)
\(L(2)\) 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 \( 1 + (-141. - 228. i)T \)
good3 \( 1 + (2.38 - 2.38i)T - 9iT^{2} \)
5 \( 1 + 4.67T + 25T^{2} \)
7 \( 1 + (-4.02 - 4.02i)T + 49iT^{2} \)
11 \( 1 + 0.758iT - 121T^{2} \)
13 \( 1 + 22.3iT - 169T^{2} \)
17 \( 1 + (-7.71 + 7.71i)T - 289iT^{2} \)
19 \( 1 + (10.4 + 10.4i)T + 361iT^{2} \)
23 \( 1 - 25.6T + 529T^{2} \)
29 \( 1 + (-9.53 - 9.53i)T + 841iT^{2} \)
31 \( 1 + (-11.6 - 11.6i)T + 961iT^{2} \)
37 \( 1 + 4.88T + 1.36e3T^{2} \)
41 \( 1 - 66.2T + 1.68e3T^{2} \)
43 \( 1 - 57.9iT - 1.84e3T^{2} \)
47 \( 1 - 47.1T + 2.20e3T^{2} \)
53 \( 1 + 45.0T + 2.80e3T^{2} \)
59 \( 1 + (-80.1 - 80.1i)T + 3.48e3iT^{2} \)
61 \( 1 + 109.T + 3.72e3T^{2} \)
67 \( 1 - 52.7T + 4.48e3T^{2} \)
71 \( 1 + (10.6 + 10.6i)T + 5.04e3iT^{2} \)
73 \( 1 - 52.6iT - 5.32e3T^{2} \)
79 \( 1 - 19.2iT - 6.24e3T^{2} \)
83 \( 1 + (26.6 + 26.6i)T + 6.88e3iT^{2} \)
89 \( 1 + 166. iT - 7.92e3T^{2} \)
97 \( 1 - 176. iT - 9.40e3T^{2} \)
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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

−10.81314719771237677354164412323, −9.991317034853016243240363691535, −8.892137865007743069772345236615, −8.052163742106099773466751802802, −7.36889646865755654535389676773, −5.95767790992550883330745804754, −5.20990920249904817327887803612, −4.44306139824753528687618410364, −2.91345502274632273150115574531, −0.67139004198237037588432616322, 0.75988692116627327658987536307, 1.89972569704000230973457452940, 3.75786436272804837069488695501, 4.55359739280939927539311000257, 6.10543531730672660242850312132, 7.07444596509136725792685521478, 7.65686344049280742104046899609, 8.629990768648368788454601454600, 9.634761525414367076926627005061, 10.92737903556496225607333932359

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