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Properties

Label 2-538-269.187-c2-0-27
Degree $2$
Conductor $538$
Sign $0.674 + 0.738i$
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 + (3.13 + 3.13i)3-s + 2i·4-s − 3.10·5-s − 6.26i·6-s + (1.53 − 1.53i)7-s + (2 − 2i)8-s + 10.6i·9-s + (3.10 + 3.10i)10-s − 18.9i·11-s + (−6.26 + 6.26i)12-s − 14.4i·13-s − 3.06·14-s + (−9.73 − 9.73i)15-s − 4·16-s + (0.930 + 0.930i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1.04 + 1.04i)3-s + 0.5i·4-s − 0.621·5-s − 1.04i·6-s + (0.218 − 0.218i)7-s + (0.250 − 0.250i)8-s + 1.18i·9-s + (0.310 + 0.310i)10-s − 1.72i·11-s + (−0.522 + 0.522i)12-s − 1.10i·13-s − 0.218·14-s + (−0.649 − 0.649i)15-s − 0.250·16-s + (0.0547 + 0.0547i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.674 + 0.738i$
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: \((2,\ 538,\ (\ :1),\ 0.674 + 0.738i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.675907587\)
\(L(\frac12)\) \(\approx\) \(1.675907587\)
\(L(2)\) 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 \( 1 + (-97.9 + 250. i)T \)
good3 \( 1 + (-3.13 - 3.13i)T + 9iT^{2} \)
5 \( 1 + 3.10T + 25T^{2} \)
7 \( 1 + (-1.53 + 1.53i)T - 49iT^{2} \)
11 \( 1 + 18.9iT - 121T^{2} \)
13 \( 1 + 14.4iT - 169T^{2} \)
17 \( 1 + (-0.930 - 0.930i)T + 289iT^{2} \)
19 \( 1 + (-12.0 + 12.0i)T - 361iT^{2} \)
23 \( 1 - 33.5T + 529T^{2} \)
29 \( 1 + (14.4 - 14.4i)T - 841iT^{2} \)
31 \( 1 + (-27.4 + 27.4i)T - 961iT^{2} \)
37 \( 1 + 51.5T + 1.36e3T^{2} \)
41 \( 1 - 18.4T + 1.68e3T^{2} \)
43 \( 1 - 41.3iT - 1.84e3T^{2} \)
47 \( 1 - 73.3T + 2.20e3T^{2} \)
53 \( 1 - 32.0T + 2.80e3T^{2} \)
59 \( 1 + (-32.0 + 32.0i)T - 3.48e3iT^{2} \)
61 \( 1 + 35.3T + 3.72e3T^{2} \)
67 \( 1 + 76.5T + 4.48e3T^{2} \)
71 \( 1 + (9.32 - 9.32i)T - 5.04e3iT^{2} \)
73 \( 1 - 52.8iT - 5.32e3T^{2} \)
79 \( 1 + 79.4iT - 6.24e3T^{2} \)
83 \( 1 + (-77.1 + 77.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 84.5iT - 7.92e3T^{2} \)
97 \( 1 + 29.7iT - 9.40e3T^{2} \)
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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

−10.58862907509601506813165500710, −9.501156852358923944567198014088, −8.789586318616803142607108154000, −8.174428663915889170912851658163, −7.37525162627273400020908620230, −5.65964398049391510491112171782, −4.41349766319451109308314963210, −3.34128030802501250788312474639, −2.92506482783089798914960716548, −0.74249137586371558283404848791, 1.43081925862586820565266088153, 2.38242473075919721398577249538, 3.94858918170318408684334990426, 5.18261390705458828358292872019, 6.80971820369316564903122206305, 7.21417806971195392843615937314, 7.908047241743502993561719577368, 8.837136357494936680149841786433, 9.441759407641860254229333990275, 10.54711174565177684632263535576

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