Properties

Label 2-507-13.3-c1-0-20
Degree $2$
Conductor $507$
Sign $-0.189 + 0.981i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1 − 1.73i)4-s − 3.46·5-s + (0.866 − 1.5i)7-s + (−0.499 + 0.866i)9-s + (−1.73 − 3i)11-s + 1.99·12-s + (−1.73 − 2.99i)15-s + (−1.99 − 3.46i)16-s + (1.73 − 3i)19-s + (−3.46 + 5.99i)20-s + 1.73·21-s + (−3 − 5.19i)23-s + 6.99·25-s − 0.999·27-s + (−1.73 − 3i)28-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.5 − 0.866i)4-s − 1.54·5-s + (0.327 − 0.566i)7-s + (−0.166 + 0.288i)9-s + (−0.522 − 0.904i)11-s + 0.577·12-s + (−0.447 − 0.774i)15-s + (−0.499 − 0.866i)16-s + (0.397 − 0.688i)19-s + (−0.774 + 1.34i)20-s + 0.377·21-s + (−0.625 − 1.08i)23-s + 1.39·25-s − 0.192·27-s + (−0.327 − 0.566i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.662104 - 0.802377i\)
\(L(\frac12)\) \(\approx\) \(0.662104 - 0.802377i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.46T + 5T^{2} \)
7 \( 1 + (-0.866 + 1.5i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.73 + 3i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 + 3i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73T + 31T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-1.73 + 3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 + 7.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.19 - 9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-3.46 - 6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.59 - 4.5i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.85753171389801112471390005666, −9.971997031733601411386986866347, −8.791877775335714392085635969057, −7.913475248172496282031119315191, −7.21500888193816314702920118976, −5.96075348484358164203105451409, −4.78098413345896893875330632069, −3.92196046091848423994123853460, −2.68354186106727918529156031439, −0.58405599576362733940293168372, 2.03863350894057191541435727098, 3.32078692640904312512137030111, 4.15236081742670923830684182112, 5.57272890615739565633452241739, 7.18190247666674498914050602100, 7.46960392183277767811737002309, 8.227395487640500814102375932502, 9.030721016718750257432225087408, 10.49011895813016899611528236883, 11.53594137402343636277473162951

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