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Properties

Label 2-539-7.4-c1-0-30
Degree $2$
Conductor $539$
Sign $-0.991 + 0.126i$
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  + (−0.328 − 0.568i)2-s + (0.956 − 1.65i)3-s + (0.784 − 1.35i)4-s + (−1.78 − 3.09i)5-s − 1.25·6-s − 2.34·8-s + (−0.328 − 0.568i)9-s + (−1.17 + 2.02i)10-s + (0.5 − 0.866i)11-s + (−1.5 − 2.59i)12-s + 5.91·13-s − 6.82·15-s + (−0.799 − 1.38i)16-s + (−0.828 + 1.43i)17-s + (−0.215 + 0.373i)18-s + (0.740 + 1.28i)19-s + ⋯
L(s)  = 1  + (−0.232 − 0.402i)2-s + (0.552 − 0.956i)3-s + (0.392 − 0.679i)4-s + (−0.798 − 1.38i)5-s − 0.512·6-s − 0.828·8-s + (−0.109 − 0.189i)9-s + (−0.370 + 0.641i)10-s + (0.150 − 0.261i)11-s + (−0.433 − 0.749i)12-s + 1.63·13-s − 1.76·15-s + (−0.199 − 0.346i)16-s + (−0.200 + 0.347i)17-s + (−0.0508 + 0.0880i)18-s + (0.169 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0927210 - 1.46109i\)
\(L(\frac12)\) \(\approx\) \(0.0927210 - 1.46109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.328 + 0.568i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.956 + 1.65i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.78 + 3.09i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.91T + 13T^{2} \)
17 \( 1 + (0.828 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.740 - 1.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (3.54 - 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.28T + 41T^{2} \)
43 \( 1 - 1.59T + 43T^{2} \)
47 \( 1 + (0.828 + 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.61 - 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.42 + 7.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.61T + 71T^{2} \)
73 \( 1 + (-2.28 + 3.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.19 - 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.167T + 83T^{2} \)
89 \( 1 + (1.28 + 2.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.73T + 97T^{2} \)
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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

−10.56677995142806614459267791794, −9.265561797270787089984114098514, −8.504028148847662199426276049785, −8.066846801192159915774050801244, −6.78493079499762236137016192764, −5.88905876425239979999924997078, −4.64453083569292857035849103626, −3.33730545640554719815425130813, −1.73113135232184658689250237830, −0.932946528203800514962271256918, 2.68961586697765794222926109821, 3.58683102041740062367396153933, 4.09437529977559777958220137003, 6.02859193962687767243227204383, 6.89594121314193588085678495298, 7.66542138255188764027724203810, 8.540810623137867117395009692341, 9.353172561293100734396474904071, 10.39975998791903137432463137241, 11.24859778913003490269074666345

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