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Properties

Label 2-539-49.23-c1-0-31
Degree $2$
Conductor $539$
Sign $0.558 - 0.829i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
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  + (−0.102 + 1.36i)2-s + (2.76 − 0.852i)3-s + (0.116 + 0.0174i)4-s + (1.55 + 1.44i)5-s + (0.883 + 3.86i)6-s + (−0.773 − 2.53i)7-s + (−0.646 + 2.83i)8-s + (4.43 − 3.02i)9-s + (−2.13 + 1.98i)10-s + (−0.826 − 0.563i)11-s + (0.335 − 0.0505i)12-s + (−1.36 + 0.656i)13-s + (3.54 − 0.799i)14-s + (5.53 + 2.66i)15-s + (−3.58 − 1.10i)16-s + (0.527 − 1.34i)17-s + ⋯
L(s)  = 1  + (−0.0725 + 0.967i)2-s + (1.59 − 0.492i)3-s + (0.0580 + 0.00874i)4-s + (0.696 + 0.646i)5-s + (0.360 + 1.57i)6-s + (−0.292 − 0.956i)7-s + (−0.228 + 1.00i)8-s + (1.47 − 1.00i)9-s + (−0.675 + 0.626i)10-s + (−0.249 − 0.169i)11-s + (0.0969 − 0.0146i)12-s + (−0.378 + 0.182i)13-s + (0.946 − 0.213i)14-s + (1.42 + 0.688i)15-s + (−0.896 − 0.276i)16-s + (0.127 − 0.325i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.558 - 0.829i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34689 + 1.24940i\)
\(L(\frac12)\) \(\approx\) \(2.34689 + 1.24940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.773 + 2.53i)T \)
11 \( 1 + (0.826 + 0.563i)T \)
good2 \( 1 + (0.102 - 1.36i)T + (-1.97 - 0.298i)T^{2} \)
3 \( 1 + (-2.76 + 0.852i)T + (2.47 - 1.68i)T^{2} \)
5 \( 1 + (-1.55 - 1.44i)T + (0.373 + 4.98i)T^{2} \)
13 \( 1 + (1.36 - 0.656i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.527 + 1.34i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (3.26 - 5.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.47 + 6.31i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-0.505 - 0.633i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-1.29 - 2.24i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.72 - 0.862i)T + (35.3 - 10.9i)T^{2} \)
41 \( 1 + (-1.48 + 6.52i)T + (-36.9 - 17.7i)T^{2} \)
43 \( 1 + (-0.735 - 3.22i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-0.770 + 10.2i)T + (-46.4 - 7.00i)T^{2} \)
53 \( 1 + (-13.9 - 2.10i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (5.86 - 5.44i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (-2.76 + 0.417i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (-2.68 - 4.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.96 - 9.98i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.440 - 5.87i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (0.534 - 0.926i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.9 + 7.19i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (4.41 - 3.01i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 - 5.69T + 97T^{2} \)
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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

−10.44916235683224009431253365008, −10.09704433091103987495569735692, −8.749879205686074062410830591911, −8.219395770221889041212241419672, −7.20853063757461491778104705797, −6.84605532003767415006236104016, −5.82095620551072659529382629972, −4.10688879909752194114047396496, −2.86228361227635667385602327444, −2.01874758278761041607571814314, 1.83127087101251984765349557308, 2.54909522775130035647363354755, 3.44645709382788197257667985779, 4.69207264996818731669878843166, 5.95301888298138656873070042355, 7.32558750105700857381907280834, 8.458633267850090590221077458279, 9.252209973823493606140763755424, 9.598341938241693620337232538391, 10.40338848386698225695203308526

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