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Properties

Label 2-538-269.187-c2-0-44
Degree $2$
Conductor $538$
Sign $-0.949 - 0.313i$
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 + (−3.11 − 3.11i)3-s + 2i·4-s − 4.51·5-s − 6.23i·6-s + (6.48 − 6.48i)7-s + (−2 + 2i)8-s + 10.4i·9-s + (−4.51 − 4.51i)10-s − 14.0i·11-s + (6.23 − 6.23i)12-s + 8.00i·13-s + 12.9·14-s + (14.0 + 14.0i)15-s − 4·16-s + (−7.95 − 7.95i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−1.03 − 1.03i)3-s + 0.5i·4-s − 0.902·5-s − 1.03i·6-s + (0.926 − 0.926i)7-s + (−0.250 + 0.250i)8-s + 1.16i·9-s + (−0.451 − 0.451i)10-s − 1.27i·11-s + (0.519 − 0.519i)12-s + 0.615i·13-s + 0.926·14-s + (0.937 + 0.937i)15-s − 0.250·16-s + (−0.467 − 0.467i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.949 - 0.313i$
Analytic conductor: \(14.6594\)
Root analytic conductor: \(3.82876\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :1),\ -0.949 - 0.313i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1643482654\)
\(L(\frac12)\) \(\approx\) \(0.1643482654\)
\(L(2)\) 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 \( 1 + (37.0 + 266. i)T \)
good3 \( 1 + (3.11 + 3.11i)T + 9iT^{2} \)
5 \( 1 + 4.51T + 25T^{2} \)
7 \( 1 + (-6.48 + 6.48i)T - 49iT^{2} \)
11 \( 1 + 14.0iT - 121T^{2} \)
13 \( 1 - 8.00iT - 169T^{2} \)
17 \( 1 + (7.95 + 7.95i)T + 289iT^{2} \)
19 \( 1 + (10.9 - 10.9i)T - 361iT^{2} \)
23 \( 1 - 26.8T + 529T^{2} \)
29 \( 1 + (25.8 - 25.8i)T - 841iT^{2} \)
31 \( 1 + (2.26 - 2.26i)T - 961iT^{2} \)
37 \( 1 + 35.4T + 1.36e3T^{2} \)
41 \( 1 + 15.7T + 1.68e3T^{2} \)
43 \( 1 - 22.2iT - 1.84e3T^{2} \)
47 \( 1 + 69.1T + 2.20e3T^{2} \)
53 \( 1 - 11.9T + 2.80e3T^{2} \)
59 \( 1 + (-0.476 + 0.476i)T - 3.48e3iT^{2} \)
61 \( 1 + 34.6T + 3.72e3T^{2} \)
67 \( 1 + 24.6T + 4.48e3T^{2} \)
71 \( 1 + (49.0 - 49.0i)T - 5.04e3iT^{2} \)
73 \( 1 - 124. iT - 5.32e3T^{2} \)
79 \( 1 - 2.34iT - 6.24e3T^{2} \)
83 \( 1 + (-97.2 + 97.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 32.5iT - 7.92e3T^{2} \)
97 \( 1 - 10.7iT - 9.40e3T^{2} \)
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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

−10.70203804341079376949843174478, −8.827156428724731481267293564467, −7.960998168777010272428316997148, −7.22439551082361386296401214543, −6.60399925027292575001231671638, −5.52111448251616043816667227786, −4.59121365131176680681313111197, −3.50677633026203062444842724018, −1.43774758160925064884810778783, −0.06375603070757601796242963025, 2.06426906840465867907230511490, 3.68086584643481493446465435513, 4.71897188019433589999820249760, 5.04146472364835833360756364850, 6.16647001508779266057985492429, 7.46849127638012641325426954181, 8.626158689608886783928448651908, 9.604074704018057003282473289814, 10.57612737467909058197071469720, 11.16013913150841380036814376335

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