Properties

Label 2-507-169.4-c1-0-27
Degree $2$
Conductor $507$
Sign $-0.741 + 0.671i$
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  + (1.81 + 0.0732i)2-s + (−0.278 + 0.960i)3-s + (1.30 + 0.105i)4-s + (−3.76 − 1.42i)5-s + (−0.576 + 1.72i)6-s + (−1.29 − 3.03i)7-s + (−1.24 − 0.150i)8-s + (−0.845 − 0.534i)9-s + (−6.74 − 2.87i)10-s + (−0.0893 − 0.141i)11-s + (−0.465 + 1.22i)12-s + (−2.65 + 2.43i)13-s + (−2.12 − 5.60i)14-s + (2.41 − 3.21i)15-s + (−4.83 − 0.786i)16-s + (−1.24 + 0.531i)17-s + ⋯
L(s)  = 1  + (1.28 + 0.0518i)2-s + (−0.160 + 0.554i)3-s + (0.654 + 0.0528i)4-s + (−1.68 − 0.638i)5-s + (−0.235 + 0.704i)6-s + (−0.488 − 1.14i)7-s + (−0.438 − 0.0533i)8-s + (−0.281 − 0.178i)9-s + (−2.13 − 0.908i)10-s + (−0.0269 − 0.0426i)11-s + (−0.134 + 0.354i)12-s + (−0.736 + 0.676i)13-s + (−0.568 − 1.49i)14-s + (0.624 − 0.831i)15-s + (−1.20 − 0.196i)16-s + (−0.302 + 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.741 + 0.671i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.741 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227458 - 0.590046i\)
\(L(\frac12)\) \(\approx\) \(0.227458 - 0.590046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (2.65 - 2.43i)T \)
good2 \( 1 + (-1.81 - 0.0732i)T + (1.99 + 0.160i)T^{2} \)
5 \( 1 + (3.76 + 1.42i)T + (3.74 + 3.31i)T^{2} \)
7 \( 1 + (1.29 + 3.03i)T + (-4.84 + 5.04i)T^{2} \)
11 \( 1 + (0.0893 + 0.141i)T + (-4.71 + 9.93i)T^{2} \)
17 \( 1 + (1.24 - 0.531i)T + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (-5.53 + 3.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.39 - 2.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0325 - 0.808i)T + (-28.9 - 2.33i)T^{2} \)
31 \( 1 + (4.06 + 4.59i)T + (-3.73 + 30.7i)T^{2} \)
37 \( 1 + (-4.10 + 0.838i)T + (34.0 - 14.5i)T^{2} \)
41 \( 1 + (4.95 + 1.43i)T + (34.6 + 21.9i)T^{2} \)
43 \( 1 + (-1.64 + 8.06i)T + (-39.5 - 16.8i)T^{2} \)
47 \( 1 + (0.289 - 0.199i)T + (16.6 - 43.9i)T^{2} \)
53 \( 1 + (-1.02 + 8.41i)T + (-51.4 - 12.6i)T^{2} \)
59 \( 1 + (-0.463 - 2.85i)T + (-55.9 + 18.6i)T^{2} \)
61 \( 1 + (11.7 - 8.82i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (0.455 + 5.64i)T + (-66.1 + 10.7i)T^{2} \)
71 \( 1 + (10.7 - 10.3i)T + (2.85 - 70.9i)T^{2} \)
73 \( 1 + (-1.49 + 2.84i)T + (-41.4 - 60.0i)T^{2} \)
79 \( 1 + (-2.80 - 4.06i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (-3.96 + 16.0i)T + (-73.4 - 38.5i)T^{2} \)
89 \( 1 + (1.00 + 0.578i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.48 + 3.66i)T + (19.4 + 95.0i)T^{2} \)
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

−10.96468070972111513353127128745, −9.680582398014116339242391600250, −8.854284301745028435134609341902, −7.51457023258983551077723925363, −6.95444463902819550651960498225, −5.45562581014995490732002256464, −4.51796206881609620884351108415, −4.01462701895301885738554139838, −3.22804110497349582271114812582, −0.24350924442075922396181944292, 2.75574634034174816584484658123, 3.32416509898651315699919345113, 4.57303822865117311442874969699, 5.57647350020076195654274124276, 6.50858568428679463578932717368, 7.47938761187194173456414770258, 8.268193772960489261216050853941, 9.445144391624895222907275101946, 10.85729884908871638402168984086, 11.69755233755891474117053560603

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