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Properties

Label 2-539-539.461-c1-0-39
Degree $2$
Conductor $539$
Sign $-0.970 + 0.241i$
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  + (0.627 − 1.30i)2-s + (−1.71 + 0.391i)3-s + (−0.0554 − 0.0694i)4-s + (−2.96 + 0.677i)5-s + (−0.566 + 2.48i)6-s + (2.62 − 0.300i)7-s + (2.69 − 0.614i)8-s + (0.0918 − 0.0442i)9-s + (−0.978 + 4.28i)10-s + (0.160 − 3.31i)11-s + (0.122 + 0.0975i)12-s + (−5.81 − 2.80i)13-s + (1.25 − 3.61i)14-s + (4.82 − 2.32i)15-s + (0.927 − 4.06i)16-s + (0.153 − 0.192i)17-s + ⋯
L(s)  = 1  + (0.443 − 0.920i)2-s + (−0.991 + 0.226i)3-s + (−0.0277 − 0.0347i)4-s + (−1.32 + 0.302i)5-s + (−0.231 + 1.01i)6-s + (0.993 − 0.113i)7-s + (0.952 − 0.217i)8-s + (0.0306 − 0.0147i)9-s + (−0.309 + 1.35i)10-s + (0.0484 − 0.998i)11-s + (0.0353 + 0.0281i)12-s + (−1.61 − 0.776i)13-s + (0.335 − 0.965i)14-s + (1.24 − 0.600i)15-s + (0.231 − 1.01i)16-s + (0.0371 − 0.0466i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0733627 - 0.599524i\)
\(L(\frac12)\) \(\approx\) \(0.0733627 - 0.599524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 + 0.300i)T \)
11 \( 1 + (-0.160 + 3.31i)T \)
good2 \( 1 + (-0.627 + 1.30i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (1.71 - 0.391i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (2.96 - 0.677i)T + (4.50 - 2.16i)T^{2} \)
13 \( 1 + (5.81 + 2.80i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-0.153 + 0.192i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + 2.60T + 19T^{2} \)
23 \( 1 + (3.91 + 4.91i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (1.98 + 1.58i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + 1.34iT - 31T^{2} \)
37 \( 1 + (7.26 - 9.11i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + (1.23 + 5.42i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-0.148 - 0.0339i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-3.36 + 6.98i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (5.82 + 7.30i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-2.82 - 0.645i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.63 - 4.56i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + 3.72T + 67T^{2} \)
71 \( 1 + (-5.11 - 6.41i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-12.0 + 5.82i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + 0.895iT - 79T^{2} \)
83 \( 1 + (-3.31 + 1.59i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-0.199 - 0.413i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 - 1.62iT - 97T^{2} \)
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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

−10.70003654803936613696132976555, −10.25528567228523452658301334364, −8.341369176493628627804648076143, −7.83648949970017960435038585946, −6.79648676689374518112250387556, −5.32242361944737940837704890786, −4.59965101410222365753097942329, −3.64499249604454303062778847821, −2.43165405565317434819996927769, −0.33186430181763000519371340311, 1.83558163646912321141935563219, 4.21209771154432150235936623011, 4.81473797714606730956935635815, 5.56565739889817555774011036429, 6.80957859415591858757829604803, 7.43333892937716403347703181325, 8.044085341991619333847958567882, 9.377752827789316805490410123515, 10.70686989158168823213378391894, 11.35642443557675819145929833274

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