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Properties

Label 2-539-49.44-c1-0-7
Degree $2$
Conductor $539$
Sign $0.882 - 0.470i$
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  + (−1.49 + 0.226i)2-s + (−1.10 − 0.751i)3-s + (0.286 − 0.0882i)4-s + (−0.220 + 2.94i)5-s + (1.82 + 0.877i)6-s + (−1.85 − 1.88i)7-s + (2.32 − 1.11i)8-s + (−0.446 − 1.13i)9-s + (−0.334 − 4.45i)10-s + (−0.365 + 0.930i)11-s + (−0.381 − 0.117i)12-s + (−2.12 − 2.66i)13-s + (3.21 + 2.40i)14-s + (2.45 − 3.07i)15-s + (−3.72 + 2.54i)16-s + (1.70 − 1.57i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.159i)2-s + (−0.636 − 0.433i)3-s + (0.143 − 0.0441i)4-s + (−0.0985 + 1.31i)5-s + (0.743 + 0.358i)6-s + (−0.702 − 0.711i)7-s + (0.821 − 0.395i)8-s + (−0.148 − 0.379i)9-s + (−0.105 − 1.41i)10-s + (−0.110 + 0.280i)11-s + (−0.110 − 0.0339i)12-s + (−0.588 − 0.737i)13-s + (0.858 + 0.642i)14-s + (0.633 − 0.793i)15-s + (−0.931 + 0.635i)16-s + (0.412 − 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.882 - 0.470i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431942 + 0.107915i\)
\(L(\frac12)\) \(\approx\) \(0.431942 + 0.107915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.85 + 1.88i)T \)
11 \( 1 + (0.365 - 0.930i)T \)
good2 \( 1 + (1.49 - 0.226i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (1.10 + 0.751i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.220 - 2.94i)T + (-4.94 - 0.745i)T^{2} \)
13 \( 1 + (2.12 + 2.66i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-1.70 + 1.57i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (1.67 - 2.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.16 - 3.86i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (0.786 + 3.44i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.186 + 0.323i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.56 - 2.95i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-3.99 + 1.92i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-8.02 - 3.86i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.31 + 0.198i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-5.61 + 1.73i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.293 + 3.91i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-0.765 - 0.236i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-2.55 - 4.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.26 - 9.93i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.06 - 1.36i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (4.26 - 7.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.438 - 0.550i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.24 + 3.16i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−10.73853212534343656720316068147, −9.931143417510923014357088254304, −9.429883849821529771298846244922, −7.935199769138892638015808333493, −7.28740428254938555815899712666, −6.72278244515551610176122096254, −5.70074750133497509750340223938, −4.03128068428231685081802026826, −2.85714815975069662998815739433, −0.798982518649777242911564314242, 0.63660693610614108791448413756, 2.35409699226644045455268729918, 4.39999743147074599747547400349, 5.04950332919231726578052013395, 6.00421459338661348383621658991, 7.42291885933599210214813747427, 8.528765163205952604870223201566, 9.024049565114452403117132624686, 9.669001955786524300568122593423, 10.64834266541292175715432121641

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