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Properties

Label 2-538-269.268-c3-0-9
Degree $2$
Conductor $538$
Sign $-0.287 + 0.957i$
Analytic cond. $31.7430$
Root an. cond. $5.63409$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 4.97i·3-s − 4·4-s + 0.491·5-s − 9.94·6-s + 17.1i·7-s − 8i·8-s + 2.28·9-s + 0.982i·10-s − 19.6·11-s − 19.8i·12-s + 14.1·13-s − 34.3·14-s + 2.44i·15-s + 16·16-s + 109. i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.956i·3-s − 0.5·4-s + 0.0439·5-s − 0.676·6-s + 0.928i·7-s − 0.353i·8-s + 0.0847·9-s + 0.0310i·10-s − 0.537·11-s − 0.478i·12-s + 0.302·13-s − 0.656·14-s + 0.0420i·15-s + 0.250·16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(31.7430\)
Root analytic conductor: \(5.63409\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{538} (537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 538,\ (\ :3/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8641537961\)
\(L(\frac12)\) \(\approx\) \(0.8641537961\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
269 \( 1 + (-4.22e3 - 1.26e3i)T \)
good3 \( 1 - 4.97iT - 27T^{2} \)
5 \( 1 - 0.491T + 125T^{2} \)
7 \( 1 - 17.1iT - 343T^{2} \)
11 \( 1 + 19.6T + 1.33e3T^{2} \)
13 \( 1 - 14.1T + 2.19e3T^{2} \)
17 \( 1 - 109. iT - 4.91e3T^{2} \)
19 \( 1 - 17.2iT - 6.85e3T^{2} \)
23 \( 1 + 168.T + 1.21e4T^{2} \)
29 \( 1 + 23.6iT - 2.43e4T^{2} \)
31 \( 1 + 259. iT - 2.97e4T^{2} \)
37 \( 1 + 54.3T + 5.06e4T^{2} \)
41 \( 1 + 165.T + 6.89e4T^{2} \)
43 \( 1 - 549.T + 7.95e4T^{2} \)
47 \( 1 + 236.T + 1.03e5T^{2} \)
53 \( 1 - 167.T + 1.48e5T^{2} \)
59 \( 1 + 254. iT - 2.05e5T^{2} \)
61 \( 1 + 653.T + 2.26e5T^{2} \)
67 \( 1 + 24.6T + 3.00e5T^{2} \)
71 \( 1 - 8.37iT - 3.57e5T^{2} \)
73 \( 1 + 945.T + 3.89e5T^{2} \)
79 \( 1 - 647.T + 4.93e5T^{2} \)
83 \( 1 - 260. iT - 5.71e5T^{2} \)
89 \( 1 - 1.53e3T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3T + 9.12e5T^{2} \)
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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

−10.74377575547470600122271671340, −10.01975725723797436928873743455, −9.288201623243672327022206934609, −8.347454607001868708430873004845, −7.65166306701511042804873983471, −6.08436261817006023596520844134, −5.71436276614402126901880111008, −4.43037682403953237778494511403, −3.68644642635924203295014688807, −2.01730414677335307769861466227, 0.26547394248870898317680968000, 1.36845811664227880518435566825, 2.53169242482219050049555341723, 3.82965507329947990053195840754, 4.90430740226397365569572278189, 6.18912881568094241412287311527, 7.30969331719629885762126185230, 7.77465153687937689946692246470, 8.984410439129819087544261285540, 10.03253994552715879583273893012

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