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Properties

Label 2-544-68.11-c1-0-8
Degree $2$
Conductor $544$
Sign $0.204 + 0.978i$
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  + (−2.46 + 0.490i)3-s + (1.38 + 0.923i)5-s + (−1.49 − 2.23i)7-s + (3.06 − 1.26i)9-s + (−0.183 + 0.923i)11-s + (−2.08 + 2.08i)13-s + (−3.86 − 1.59i)15-s + (−1.82 − 3.69i)17-s + (1.14 − 2.75i)19-s + (4.76 + 4.76i)21-s + (3.87 + 0.771i)23-s + (−0.855 − 2.06i)25-s + (−0.662 + 0.442i)27-s + (2.89 − 4.33i)29-s + (−1.60 − 8.04i)31-s + ⋯
L(s)  = 1  + (−1.42 + 0.283i)3-s + (0.618 + 0.413i)5-s + (−0.563 − 0.843i)7-s + (1.02 − 0.423i)9-s + (−0.0554 + 0.278i)11-s + (−0.577 + 0.577i)13-s + (−0.997 − 0.412i)15-s + (−0.443 − 0.896i)17-s + (0.261 − 0.631i)19-s + (1.04 + 1.04i)21-s + (0.808 + 0.160i)23-s + (−0.171 − 0.412i)25-s + (−0.127 + 0.0851i)27-s + (0.538 − 0.805i)29-s + (−0.287 − 1.44i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.204 + 0.978i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487837 - 0.396247i\)
\(L(\frac12)\) \(\approx\) \(0.487837 - 0.396247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (1.82 + 3.69i)T \)
good3 \( 1 + (2.46 - 0.490i)T + (2.77 - 1.14i)T^{2} \)
5 \( 1 + (-1.38 - 0.923i)T + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (1.49 + 2.23i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (0.183 - 0.923i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 13iT^{2} \)
19 \( 1 + (-1.14 + 2.75i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-3.87 - 0.771i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-2.89 + 4.33i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (1.60 + 8.04i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (1.66 + 8.39i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-5.29 + 3.53i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (-4.80 - 11.6i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (5.14 + 5.14i)T + 47iT^{2} \)
53 \( 1 + (3.98 + 1.65i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.68 - 0.699i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.26 + 3.38i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + 6.72T + 67T^{2} \)
71 \( 1 + (-7.39 + 1.47i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (-0.680 - 0.454i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.644 + 3.24i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (9.49 + 3.93i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-12.5 - 12.5i)T + 89iT^{2} \)
97 \( 1 + (8.63 - 12.9i)T + (-37.1 - 89.6i)T^{2} \)
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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

−10.75796082284196930150836923420, −9.785971306889662951589275488711, −9.382365005015340489327917525858, −7.53433048810012304335072851457, −6.77105675241537864005891607462, −6.09545334737776569330631886967, −5.01718108058445849873081169420, −4.18529818357859033328379387598, −2.51217393190410858764629103059, −0.45675189385435691979106734269, 1.39354814526775250191290865448, 3.05228772426949449773351212898, 4.85075953173193960200851636906, 5.57036257517181820666937061988, 6.18321256597125451125162722880, 7.08164166562885234531180530472, 8.452125031297631614249600572871, 9.312387592685277149262774683347, 10.32646064094050007694387087591, 10.94378920031445058338104082106

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