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Properties

Label 2-544-544.77-c1-0-20
Degree $2$
Conductor $544$
Sign $0.629 - 0.776i$
Analytic cond. $4.34386$
Root an. cond. $2.08419$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.37 + 0.350i)2-s + (1.88 − 1.88i)3-s + (1.75 − 0.960i)4-s + 4.17i·5-s + (−1.92 + 3.24i)6-s + (−0.584 − 0.242i)7-s + (−2.06 + 1.93i)8-s − 4.09i·9-s + (−1.46 − 5.71i)10-s + (2.20 + 2.20i)11-s + (1.49 − 5.11i)12-s + (−1.68 + 4.07i)13-s + (0.885 + 0.126i)14-s + (7.86 + 7.86i)15-s + (2.15 − 3.36i)16-s + (4.03 + 0.869i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.247i)2-s + (1.08 − 1.08i)3-s + (0.877 − 0.480i)4-s + 1.86i·5-s + (−0.784 + 1.32i)6-s + (−0.220 − 0.0914i)7-s + (−0.730 + 0.682i)8-s − 1.36i·9-s + (−0.462 − 1.80i)10-s + (0.665 + 0.665i)11-s + (0.431 − 1.47i)12-s + (−0.468 + 1.13i)13-s + (0.236 + 0.0339i)14-s + (2.02 + 2.02i)15-s + (0.538 − 0.842i)16-s + (0.977 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.629 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16245 + 0.554149i\)
\(L(\frac12)\) \(\approx\) \(1.16245 + 0.554149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.350i)T \)
17 \( 1 + (-4.03 - 0.869i)T \)
good3 \( 1 + (-1.88 + 1.88i)T - 3iT^{2} \)
5 \( 1 - 4.17iT - 5T^{2} \)
7 \( 1 + (0.584 + 0.242i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.20 - 2.20i)T + 11iT^{2} \)
13 \( 1 + (1.68 - 4.07i)T + (-9.19 - 9.19i)T^{2} \)
19 \( 1 + (5.12 - 2.12i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-1.07 - 0.447i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.43 + 3.43i)T + 29iT^{2} \)
31 \( 1 + (-9.82 - 4.06i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + 3.39T + 37T^{2} \)
41 \( 1 + (-2.12 + 5.13i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (-2.33 + 5.63i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.96T + 47T^{2} \)
53 \( 1 + (-5.02 - 2.07i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.91 + 0.793i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + 3.64T + 61T^{2} \)
67 \( 1 + (-0.318 + 0.768i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-2.44 + 1.01i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-14.4 + 5.98i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.65 + 4.00i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (9.61 - 3.98i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-2.20 - 2.20i)T + 89iT^{2} \)
97 \( 1 + (1.60 - 3.86i)T + (-68.5 - 68.5i)T^{2} \)
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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

−10.63929295570880811660950307641, −9.964244411345793126502121871141, −9.098336572008704536811064093582, −8.071823898003161776175062030255, −7.26630926257603085369163379427, −6.83273943040293888651520916195, −6.17675193865764939808503164293, −3.70936427112552262726782276287, −2.54477189296396808119623387358, −1.84447277582214527614547004656, 0.931105853370967528263788174032, 2.67205213772351776891261857250, 3.77746876507431959014315585116, 4.79997738478089974885282509226, 6.00591651629590959105379702456, 7.77053596797984439107691938326, 8.408851015371432261263434516451, 8.937153561621774127905482702450, 9.606526904537298936755366599279, 10.20935925752351799517732199622

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