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Properties

Label 2-538-269.82-c2-0-40
Degree $2$
Conductor $538$
Sign $-0.466 + 0.884i$
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 + (1.93 − 1.93i)3-s − 2i·4-s + 9.46·5-s − 3.87i·6-s + (−8.10 − 8.10i)7-s + (−2 − 2i)8-s + 1.51i·9-s + (9.46 − 9.46i)10-s − 1.78i·11-s + (−3.87 − 3.87i)12-s − 5.85i·13-s − 16.2·14-s + (18.3 − 18.3i)15-s − 4·16-s + (−3.08 + 3.08i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.645 − 0.645i)3-s − 0.5i·4-s + 1.89·5-s − 0.645i·6-s + (−1.15 − 1.15i)7-s + (−0.250 − 0.250i)8-s + 0.167i·9-s + (0.946 − 0.946i)10-s − 0.162i·11-s + (−0.322 − 0.322i)12-s − 0.450i·13-s − 1.15·14-s + (1.22 − 1.22i)15-s − 0.250·16-s + (−0.181 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.466 + 0.884i$
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: \((2,\ 538,\ (\ :1),\ -0.466 + 0.884i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.387572909\)
\(L(\frac12)\) \(\approx\) \(3.387572909\)
\(L(2)\) 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 \( 1 + (-157. - 217. i)T \)
good3 \( 1 + (-1.93 + 1.93i)T - 9iT^{2} \)
5 \( 1 - 9.46T + 25T^{2} \)
7 \( 1 + (8.10 + 8.10i)T + 49iT^{2} \)
11 \( 1 + 1.78iT - 121T^{2} \)
13 \( 1 + 5.85iT - 169T^{2} \)
17 \( 1 + (3.08 - 3.08i)T - 289iT^{2} \)
19 \( 1 + (13.1 + 13.1i)T + 361iT^{2} \)
23 \( 1 + 2.18T + 529T^{2} \)
29 \( 1 + (1.77 + 1.77i)T + 841iT^{2} \)
31 \( 1 + (-5.27 - 5.27i)T + 961iT^{2} \)
37 \( 1 - 11.7T + 1.36e3T^{2} \)
41 \( 1 - 38.1T + 1.68e3T^{2} \)
43 \( 1 - 61.6iT - 1.84e3T^{2} \)
47 \( 1 - 65.9T + 2.20e3T^{2} \)
53 \( 1 - 82.5T + 2.80e3T^{2} \)
59 \( 1 + (49.0 + 49.0i)T + 3.48e3iT^{2} \)
61 \( 1 + 97.6T + 3.72e3T^{2} \)
67 \( 1 + 33.0T + 4.48e3T^{2} \)
71 \( 1 + (-18.4 - 18.4i)T + 5.04e3iT^{2} \)
73 \( 1 - 104. iT - 5.32e3T^{2} \)
79 \( 1 + 138. iT - 6.24e3T^{2} \)
83 \( 1 + (106. + 106. i)T + 6.88e3iT^{2} \)
89 \( 1 - 42.1iT - 7.92e3T^{2} \)
97 \( 1 - 100. iT - 9.40e3T^{2} \)
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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

−10.39523494378001859674289113302, −9.590794044502627811734011664673, −8.828867778965350308167185590342, −7.40209503407748429509490275753, −6.51347833284497536234272592415, −5.84568758280971946466050486191, −4.54173775205652796755190687255, −3.06436578937333718417239766242, −2.30827360156037364742093200806, −1.06483716049526429949223088122, 2.20183305945562029290432453938, 2.95669936021456496093109886469, 4.26538232615527459021314671958, 5.66232722200713744442415811442, 6.04708995395022883104058874577, 6.92372720470021171268325220680, 8.693585160311451596000748609801, 9.142107024395618751099674809220, 9.742583247087184806664439387649, 10.52640504377591832909349485179

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