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Properties

Label 2-538-269.82-c2-0-14
Degree $2$
Conductor $538$
Sign $-0.00709 - 0.999i$
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 + (1.35 − 1.35i)3-s − 2i·4-s + 4.13·5-s + 2.70i·6-s + (1.20 + 1.20i)7-s + (2 + 2i)8-s + 5.32i·9-s + (−4.13 + 4.13i)10-s + 19.3i·11-s + (−2.70 − 2.70i)12-s − 6.76i·13-s − 2.41·14-s + (5.60 − 5.60i)15-s − 4·16-s + (−16.6 + 16.6i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.451 − 0.451i)3-s − 0.5i·4-s + 0.826·5-s + 0.451i·6-s + (0.172 + 0.172i)7-s + (0.250 + 0.250i)8-s + 0.592i·9-s + (−0.413 + 0.413i)10-s + 1.75i·11-s + (−0.225 − 0.225i)12-s − 0.520i·13-s − 0.172·14-s + (0.373 − 0.373i)15-s − 0.250·16-s + (−0.980 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00709 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-0.00709 - 0.999i$
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: \((2,\ 538,\ (\ :1),\ -0.00709 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.613220650\)
\(L(\frac12)\) \(\approx\) \(1.613220650\)
\(L(2)\) 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 \( 1 + (-242. - 117. i)T \)
good3 \( 1 + (-1.35 + 1.35i)T - 9iT^{2} \)
5 \( 1 - 4.13T + 25T^{2} \)
7 \( 1 + (-1.20 - 1.20i)T + 49iT^{2} \)
11 \( 1 - 19.3iT - 121T^{2} \)
13 \( 1 + 6.76iT - 169T^{2} \)
17 \( 1 + (16.6 - 16.6i)T - 289iT^{2} \)
19 \( 1 + (-1.42 - 1.42i)T + 361iT^{2} \)
23 \( 1 + 16.5T + 529T^{2} \)
29 \( 1 + (-15.5 - 15.5i)T + 841iT^{2} \)
31 \( 1 + (4.83 + 4.83i)T + 961iT^{2} \)
37 \( 1 - 70.2T + 1.36e3T^{2} \)
41 \( 1 + 70.2T + 1.68e3T^{2} \)
43 \( 1 - 8.53iT - 1.84e3T^{2} \)
47 \( 1 - 42.1T + 2.20e3T^{2} \)
53 \( 1 + 14.6T + 2.80e3T^{2} \)
59 \( 1 + (-73.5 - 73.5i)T + 3.48e3iT^{2} \)
61 \( 1 - 9.57T + 3.72e3T^{2} \)
67 \( 1 - 46.8T + 4.48e3T^{2} \)
71 \( 1 + (55.1 + 55.1i)T + 5.04e3iT^{2} \)
73 \( 1 - 140. iT - 5.32e3T^{2} \)
79 \( 1 + 138. iT - 6.24e3T^{2} \)
83 \( 1 + (31.5 + 31.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 90.2iT - 7.92e3T^{2} \)
97 \( 1 - 70.9iT - 9.40e3T^{2} \)
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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

−10.36383279682284346408226401754, −10.01426704443752341005882333466, −8.940071359209233196319027526329, −8.125095800053820195640163983413, −7.33683866355807811918215378748, −6.45296453024541436511604139830, −5.41547860753927204733200502883, −4.38191023923130499142904228412, −2.35004758233802669134358060412, −1.70795927638412764206993037959, 0.69214441311686467217315960465, 2.30977433649205079342743479340, 3.34942442999658786577398306506, 4.42565541050867798601581146948, 5.86464113482176289372654504239, 6.70144601824522740602123289860, 8.100149263160922657960629867512, 8.860556626934470182907160230027, 9.493071738832776871235856621564, 10.20314143669211838358534982796

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