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Properties

Label 2-539-7.2-c1-0-31
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  + (1.24 − 2.15i)2-s + (−0.356 − 0.617i)3-s + (−2.10 − 3.64i)4-s + (1.10 − 1.90i)5-s − 1.77·6-s − 5.49·8-s + (1.24 − 2.15i)9-s + (−2.74 − 4.75i)10-s + (0.5 + 0.866i)11-s + (−1.5 + 2.59i)12-s + 3.28·13-s − 1.57·15-s + (−2.63 + 4.56i)16-s + (0.745 + 1.29i)17-s + (−3.10 − 5.37i)18-s + (−3.45 + 5.99i)19-s + ⋯
L(s)  = 1  + (0.880 − 1.52i)2-s + (−0.205 − 0.356i)3-s + (−1.05 − 1.82i)4-s + (0.492 − 0.853i)5-s − 0.725·6-s − 1.94·8-s + (0.415 − 0.719i)9-s + (−0.868 − 1.50i)10-s + (0.150 + 0.261i)11-s + (−0.433 + 0.749i)12-s + 0.911·13-s − 0.406·15-s + (−0.658 + 1.14i)16-s + (0.180 + 0.313i)17-s + (−0.731 − 1.26i)18-s + (−0.793 + 1.37i)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: $\chi_{539} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.140423 + 2.21277i\)
\(L(\frac12)\) \(\approx\) \(0.140423 + 2.21277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 + 2.15i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.356 + 0.617i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.10 + 1.90i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 3.28T + 13T^{2} \)
17 \( 1 + (-0.745 - 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.64T + 29T^{2} \)
31 \( 1 + (1.17 + 2.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.77 + 4.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 + (-0.745 + 1.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.152 + 0.263i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.32 + 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.49 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + (4.28 + 7.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.31 - 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.93T + 83T^{2} \)
89 \( 1 + (-1.60 + 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.85T + 97T^{2} \)
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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

−10.58986891429038821561878042942, −9.632458912062716206655142964056, −9.116850117101875926621861695016, −7.74361796104417537098276159624, −6.08229694504796579695401884927, −5.63454798768861762129013736986, −4.23793211186419222583129754512, −3.66370255664173488083461367337, −1.94354689416196153063327432544, −1.14719155078597771601673029818, 2.66487921947706523546499575561, 4.07196893153165186242275805482, 4.82808437595826794705611878247, 5.97845552376789882453171932005, 6.48726649750344022084732720486, 7.39366485305285224758862651724, 8.310688337391182149577174013899, 9.285916385964559171173247531373, 10.57291776319105519469918110915, 11.08322576489637716886917084588

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