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Properties

Label 2-544-136.107-c1-0-10
Degree $2$
Conductor $544$
Sign $0.992 - 0.122i$
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  + (0.456 + 0.683i)3-s + (1.61 + 0.322i)5-s + (4.21 − 0.837i)7-s + (0.889 − 2.14i)9-s + (0.106 + 0.0712i)11-s + (−2.31 + 2.31i)13-s + (0.518 + 1.25i)15-s + (−2.89 − 2.93i)17-s + (−1.68 − 4.06i)19-s + (2.49 + 2.49i)21-s + (2.14 + 1.43i)23-s + (−2.10 − 0.870i)25-s + (4.29 − 0.853i)27-s + (−0.908 + 4.56i)29-s + (2.02 + 3.02i)31-s + ⋯
L(s)  = 1  + (0.263 + 0.394i)3-s + (0.723 + 0.144i)5-s + (1.59 − 0.316i)7-s + (0.296 − 0.715i)9-s + (0.0321 + 0.0214i)11-s + (−0.642 + 0.642i)13-s + (0.134 + 0.323i)15-s + (−0.702 − 0.711i)17-s + (−0.386 − 0.932i)19-s + (0.544 + 0.544i)21-s + (0.447 + 0.299i)23-s + (−0.420 − 0.174i)25-s + (0.825 − 0.164i)27-s + (−0.168 + 0.848i)29-s + (0.363 + 0.543i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 544 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(544\)    =    \(2^{5} \cdot 17\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(4.34386\)
Root analytic conductor: \(2.08419\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{544} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 544,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02260 + 0.124477i\)
\(L(\frac12)\) \(\approx\) \(2.02260 + 0.124477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.89 + 2.93i)T \)
good3 \( 1 + (-0.456 - 0.683i)T + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-1.61 - 0.322i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-4.21 + 0.837i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.106 - 0.0712i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (2.31 - 2.31i)T - 13iT^{2} \)
19 \( 1 + (1.68 + 4.06i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-2.14 - 1.43i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.908 - 4.56i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-2.02 - 3.02i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-7.00 + 4.68i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-2.02 - 10.1i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (3.26 - 7.87i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (4.90 - 4.90i)T - 47iT^{2} \)
53 \( 1 + (1.32 - 0.549i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (12.6 + 5.25i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (0.813 + 4.09i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + 3.72iT - 67T^{2} \)
71 \( 1 + (0.759 - 0.507i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (0.631 - 3.17i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (6.49 - 9.71i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-4.75 + 1.97i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.886 - 0.886i)T - 89iT^{2} \)
97 \( 1 + (8.67 + 1.72i)T + (89.6 + 37.1i)T^{2} \)
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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

−11.01047801332821159334543997334, −9.702254561009653258186534132620, −9.290114612111768113605358336093, −8.236251843008614864896130507613, −7.20425683759261787834996849343, −6.33346857377866035717023362567, −4.85620826962217941744669975484, −4.45841409528854566222256375437, −2.79075639229521362757802058768, −1.50369638096455314117431299187, 1.67557190639961370376101294323, 2.35884279419646686757796459794, 4.30041356487531870233299258710, 5.19901964538140118056197833414, 6.07328982333636994162151056938, 7.46470457258441263578368400722, 8.086989944487229593639557547005, 8.805126903056911264075515116502, 10.05537214761789478506007423461, 10.70332342394672204124322442066

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