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Properties

Label 2-538-269.82-c2-0-2
Degree $2$
Conductor $538$
Sign $-0.217 - 0.976i$
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 + (1.87 − 1.87i)3-s − 2i·4-s − 6.42·5-s − 3.74i·6-s + (−4.44 − 4.44i)7-s + (−2 − 2i)8-s + 1.97i·9-s + (−6.42 + 6.42i)10-s + 21.9i·11-s + (−3.74 − 3.74i)12-s − 3.05i·13-s − 8.89·14-s + (−12.0 + 12.0i)15-s − 4·16-s + (−3.40 + 3.40i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.624 − 0.624i)3-s − 0.5i·4-s − 1.28·5-s − 0.624i·6-s + (−0.635 − 0.635i)7-s + (−0.250 − 0.250i)8-s + 0.219i·9-s + (−0.642 + 0.642i)10-s + 1.99i·11-s + (−0.312 − 0.312i)12-s − 0.235i·13-s − 0.635·14-s + (−0.803 + 0.803i)15-s − 0.250·16-s + (−0.200 + 0.200i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.217 - 0.976i$
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: \((2,\ 538,\ (\ :1),\ -0.217 - 0.976i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1543805941\)
\(L(\frac12)\) \(\approx\) \(0.1543805941\)
\(L(2)\) 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 \( 1 + (261. + 63.4i)T \)
good3 \( 1 + (-1.87 + 1.87i)T - 9iT^{2} \)
5 \( 1 + 6.42T + 25T^{2} \)
7 \( 1 + (4.44 + 4.44i)T + 49iT^{2} \)
11 \( 1 - 21.9iT - 121T^{2} \)
13 \( 1 + 3.05iT - 169T^{2} \)
17 \( 1 + (3.40 - 3.40i)T - 289iT^{2} \)
19 \( 1 + (22.3 + 22.3i)T + 361iT^{2} \)
23 \( 1 + 20.1T + 529T^{2} \)
29 \( 1 + (-25.9 - 25.9i)T + 841iT^{2} \)
31 \( 1 + (1.70 + 1.70i)T + 961iT^{2} \)
37 \( 1 + 33.6T + 1.36e3T^{2} \)
41 \( 1 - 24.0T + 1.68e3T^{2} \)
43 \( 1 + 2.62iT - 1.84e3T^{2} \)
47 \( 1 + 4.22T + 2.20e3T^{2} \)
53 \( 1 + 92.1T + 2.80e3T^{2} \)
59 \( 1 + (-51.3 - 51.3i)T + 3.48e3iT^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 - 36.5T + 4.48e3T^{2} \)
71 \( 1 + (94.9 + 94.9i)T + 5.04e3iT^{2} \)
73 \( 1 + 62.4iT - 5.32e3T^{2} \)
79 \( 1 - 115. iT - 6.24e3T^{2} \)
83 \( 1 + (84.2 + 84.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 31.9iT - 7.92e3T^{2} \)
97 \( 1 - 69.3iT - 9.40e3T^{2} \)
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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

−10.83653162399113079521804805612, −10.23187400766146484470288375366, −9.079084037707861480076070323942, −8.022168742853519273495883664255, −7.23036228298313399429984962936, −6.65810101739392506576743906170, −4.75675173730598663799358561256, −4.17252826262813073411252892004, −2.97169200848113165887976951293, −1.82532539071853417960791865450, 0.04459266331234178024833839031, 2.90056229511609153226074635451, 3.65290003959896647709799737360, 4.31396593215711606512940761750, 5.88734015211020937010598642600, 6.46783528428244837622263536517, 8.005828323930719953313496568362, 8.404646497203722967018320315681, 9.171787947057881556331720958437, 10.36459412046771680847879104772

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