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Properties

Label 2-538-269.82-c2-0-5
Degree $2$
Conductor $538$
Sign $0.157 - 0.987i$
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 + (0.991 − 0.991i)3-s − 2i·4-s − 6.41·5-s − 1.98i·6-s + (−6.33 − 6.33i)7-s + (−2 − 2i)8-s + 7.03i·9-s + (−6.41 + 6.41i)10-s − 6.36i·11-s + (−1.98 − 1.98i)12-s + 22.5i·13-s − 12.6·14-s + (−6.35 + 6.35i)15-s − 4·16-s + (−1.76 + 1.76i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.330 − 0.330i)3-s − 0.5i·4-s − 1.28·5-s − 0.330i·6-s + (−0.904 − 0.904i)7-s + (−0.250 − 0.250i)8-s + 0.781i·9-s + (−0.641 + 0.641i)10-s − 0.578i·11-s + (−0.165 − 0.165i)12-s + 1.73i·13-s − 0.904·14-s + (−0.423 + 0.423i)15-s − 0.250·16-s + (−0.103 + 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $0.157 - 0.987i$
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: \((2,\ 538,\ (\ :1),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5779481514\)
\(L(\frac12)\) \(\approx\) \(0.5779481514\)
\(L(2)\) 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 \( 1 + (219. + 155. i)T \)
good3 \( 1 + (-0.991 + 0.991i)T - 9iT^{2} \)
5 \( 1 + 6.41T + 25T^{2} \)
7 \( 1 + (6.33 + 6.33i)T + 49iT^{2} \)
11 \( 1 + 6.36iT - 121T^{2} \)
13 \( 1 - 22.5iT - 169T^{2} \)
17 \( 1 + (1.76 - 1.76i)T - 289iT^{2} \)
19 \( 1 + (-21.8 - 21.8i)T + 361iT^{2} \)
23 \( 1 - 10.5T + 529T^{2} \)
29 \( 1 + (15.1 + 15.1i)T + 841iT^{2} \)
31 \( 1 + (-1.42 - 1.42i)T + 961iT^{2} \)
37 \( 1 + 51.9T + 1.36e3T^{2} \)
41 \( 1 + 28.6T + 1.68e3T^{2} \)
43 \( 1 - 49.5iT - 1.84e3T^{2} \)
47 \( 1 + 57.7T + 2.20e3T^{2} \)
53 \( 1 - 20.7T + 2.80e3T^{2} \)
59 \( 1 + (38.8 + 38.8i)T + 3.48e3iT^{2} \)
61 \( 1 + 110.T + 3.72e3T^{2} \)
67 \( 1 + 80.9T + 4.48e3T^{2} \)
71 \( 1 + (-34.1 - 34.1i)T + 5.04e3iT^{2} \)
73 \( 1 + 3.81iT - 5.32e3T^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 + (-55.4 - 55.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 78.4iT - 7.92e3T^{2} \)
97 \( 1 - 42.6iT - 9.40e3T^{2} \)
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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

−11.02949414040131225628762793361, −10.09072552100877206815568427143, −9.106310004667486199987392043960, −7.974513932306546381743877707093, −7.25178561628261495919797217884, −6.37210674803740797147185008422, −4.86541051821778984855117740800, −3.85722455824050093415146676074, −3.23052924517527015891164063763, −1.54852822126656550194672512865, 0.18376825165047844610796146593, 3.11044066753158905758004133749, 3.34228715168736484937896129537, 4.74894805227220285245321342598, 5.68319146633123051734548587178, 6.88337353434862238402948879981, 7.58798490695578250026068055013, 8.650933762325230164945078133649, 9.286411958846838136418105742103, 10.36025954836195047674373975914

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